Note; the following is a combination of etymology, translation and in some cases interpretation. It also contains spoilers up through ride 47.
With regards to character's names, in the anime and manga almost every character's name has a hidden meaning, because they are written with a kanji surname and katakana given name. For the unfamiliar, Japan has three ways of writing; katakana is usually used to write non-Japanese words, like with Mr. Mark's name;
This is just squiggles to most English speakers, but let's break that down a bit. For starters, マ is read as "Ma," ー marks a long vowel, and ク is "Ku." The "·" mark separates first and last names in katakana. So we read マーク as "Maaku." An 'aa' sound tends to represent an R sound, though the only way to really tell is to hear the word pronounced. In this case it's "Mark." ホ is "Ho," ワ is "Wa," イ is "I"(sounds like eee,) テ is "Te," ン is "N" and グ is "Gu." So sound it out! Ho-wa-i-te-n-gu, which comes out in English as "Whiting."
マーク·ホワイテイング = Mark Whiting
Hiragana is used for Japanese words, so normally you could write Aichi as あ(a)い(i)ち(chi). Kanji are imported Chinese characters which can have as few as one or as many as ten distinct meanings based on how they are read--typically, tiny hiragana or katakana are placed above or next to the kanji to indicate pronunciation.
Having three writing systems may seem extraneous to English speakers, but it's actually quite useful. Katakana and hiragana are how a word "sounds"--they're phonetic alphabets. So you can show very easily how a word sounds in Japanese, because レ(re)ン(n) always sounds like "Ren" no matter what kanji is used for it. When we want to know how something sounds in English though, we have to resort to IPA, which no one outside of a graduate school English program actually understands.
For an example of how this system sees use, the Digimon Xros Wars manga had an instance where the kanji for "Melody" was written, but the katakana said to pronounce it as "Life." The result was that the word in question was both melody and life simultaneously.
However, the current situation where hiragana is for indigenous words and katakana for foreign ones is the opposite of the original system. Prior to World War II, katakana was used to write Japanese words and names where hiragana is used today. Computers from before 1990 still used katakana exclusively. And because katakana was used to write personal names, many elderly people still have katakana given names. So writing a name in katakana can make it seem older and more "historic" the way saying "foreby" instead of "beside" sounds older in English.
What I'm getting at is that Vanguard uses katakana given names and kanji surnames. It's not that unusual in anime, but this is slightly more relevant in that most of the main characters have names that tend toward historical places and events, or archaic words. Let's take a look at Aichi's name;
The kanji for his name is 先導 which reads as "Sendou" and means "leadership." An alternate reading is "Vanguard"(as in the soldiers at the front lines.) The katakana アイチ just reads "Aichi"(though on a more humorous note, my Japanese class loved to confuse the katakana チ with the kanji 千, which is 'sen'(thousand) and not 'chi.') I mentioned before that some Vanguard characters have historically-based names, so where did Aichi's come from? Well, it's the name of a prefecture in Japan, which takes its name from the original name of the tidal flats within it, Ayuchi. The Ayuchi flats are significant for being referenced in a poetry anthology, the Man'yōshū, where the poet Takechi Kurohito compares the calling of a crane to the sound of the waves in Ayuchi. These flats are still around as the Fujimae, but have largely been destroyed by cultivation and reclamation efforts that began in the Edo period.
Unlike with certain names to come(*cough* Kamui and Suzugamori I'll get to you guys in a bit), Aichi/Ayuchi isn't especially relevant to the anime on its own(see below for how it relates to Kai), outside of setting precedents for the other names, but those of you who have watched up to ride 43 or so can certainly see why the destroyed nature of the mudflats could be interpreted as relevant. Sendou of course, is especially on-topic for Vanguard, as it's the title.
Next let's look at Kai Toshiki. And before anything else, listen to the tournament announcer in ride 30; Japanese naming order is surname-forename, so Kai really is his family name.
櫂 refers to a paddle or oar, the kind you use to row a boat. Alternate meanings for Kai are "change," "the action to correct," and "ocean." Kai's surname referring to an oar where Aichi's forename references a body of water, and the first two alternate meanings for Kai, are all too painful when considering rides 40-44. Kai has inspired multiple changes in Aichi, but not all of them are necessarily for the better, and he does have to go back to correct his past mistakes in 44. Consider the following shots, from ride 33:
Aichi and Kai, both reflected in the water. Kai's appearance in said ride is shown as a ripple moving through the pool's surface, much as an oar makes ripples in the sea. We had a similar lone shot of Aichi in ride 2, minus the ripple, which shows an amazing amount of foresight and planning despite Vanguard originally being approved for just 13 episodes, as it's ride 2 that kicks off his quest for greater strength and ride 33 where he resolves it. Both of these rides use water as a character theme, relating back to Aichi and Kai's names. As an oar spurs water, Kai spurs Aichi to seek greater power, and like a ripple effect sets the stage for the series' plot.
Third, we'll take apart Tokura Misaki's name.
戸倉 contains the kanji for 'door' and 'warehouse/storehouse/treasury.' Misaki's main motif throughout the series is that of the key she lost ten years ago, which is incorporated into her summer and 44-on outfit(she wears it over her clothes in summer and beneath her blouse in winter.) It's therefor fitting that her name would etymologize as something like "door to the treasury1," especially with regards to ride 25, where Misaki finally uses that lost key to open her parents' treasure box.
葛 is the kanji for kudzu or arrowroot, 木 is wood, as in the wood element in Wu Xing. This one I had to sit on for a while, but 木 refers to the growing stage of the five Chinese elements, the "tree" stage. This wood/tree attribute is associated with strength and flexibility, as well as warmth, generosity and cooperation. Now consider that out of the core four, Kamui is the one who does most of the growing. Compare the Kamui we meed in ride 5--a lecherous, self-absorbed and overconfident boy--to the dedicated, earnest fighter who takes over in friends' matches out of concern for their health in ride 44. Kamui initially embodies his attributes the least out of anyone in the cast, being particularly weak in the cooperation and flexibility department. Kamui refuses to work together with Kai and doesn't trust his teammates' judgment, until he finally resolves to be a better fighter in ride 33.
As a note into the 'historical name' idea, 葛木 is a heterograph for both a World War II aircraft carrier and a corvette that was in service during both the Second Sino-Japanese War and the Russo-Japanese war. The aircraft carrier bit is particularly amusing, as the Katsuragi carrier never embarked with her intended aircraft, and spent the entire war being shot at while in various ports, before taking on a brief repatriation stint and being quietly retired in 1946. Thus, even more extensively than the Yamato before her, Katsuragi was quite the paper tiger; Kamui himself is built up as an incredibly strong opponent in ride 25, with 29 off-screen victories at the season's start, and yet up through ride 47 he only has 6 full fights, just 3 of which are wins.
As for Kamui's forename, the primary meaning behind カムイ are the kamuy/kamui deities worshiped by the Ainu people, the indigenous population of Japan and Russia. Kamuy were the forerunners of the kami worshiped in the Shinto religion, and were accordingly multitudinous. The Ainu religion is even more specific than Shinto in its deities' areas of expertise, with specific kamuy devoted to undertow and abstract thresholds. This divine etymology fits with Kamui's perception of himself as the one of the strongest fighters, though it's certainly more humorous than some of the other names here.
From the pool of antagonists, Suzugamori Ren;
This is where I first noticed a trend in the names. Out of the core four characters, three of them(Aichi, Misaki and Kamui) have names that relate to the "old" Japan. Ren continues this trend. His name shares its kanji with the Suzugamori execution grounds, a place historically used for the execution of criminals, conspirators against the Tokugawa Shogunate and Christians. Popular methods of execution included crucifixion and immolation at the stake. Over 100,000 people were executed at the Suzugamori grounds over the course of 220 years, so it's quite the ominous name for Vanguard's chief antagonist.
On a lighter note, remember the Megacolony player from ride 8? His name was Kishida Osamu;
His family name contains the kanji for beach and rice paddy. Japanese family names often use a location-based naming structure. Osamu's given name comes from "osamushi," ground beetle, to go with Osamu's bug theme and Megacolony deck. It's a neat touch.
Since this took a couple days to write, I'll close this post with a break down of Morikawa's name;
Morikawa Katsumi. 森川 is another location-based name, referring to a river running through a forest. Katsumi is most probably from "勝つ (katsu)," to win. Here it's intended ironically, as Morikawa has the single worst win ratio in the entire cast; 20 fights and 20 losses.
1This is really flowery and like saying that Alexander comes out as "Defender of Men" or Ariel means "Lion of God" because it's not at all an obvious meaning to a native speaker. Of course, that means exactly nothing to writers, who are wont to know as many etymologies and use them as they please.
2Red-Eye and Blue-Eye are themselves taken from the Shinto pantheon's kitsune, a type of fox spirit that fulfills many different functions based on the particular mythology behind a given tale, but primarily work as messengers of Inari, a god of rice, fertility, agriculture and industry. Inari is still worshiped today as an important deity central to everyday life.