Is being a vegan completely following halacha? Even if not eating anything traif (non-kosher), perhaps it's not truly kosher because you are not following the commandments to domesticate animals and eat meat.
Contrary to what many frum Jews believe, it is not required to eat meat on Shabbos and Yom Tov. So as far as the dietary aspect of veganism goes, it is permissible.
But many interpret veganism to mean that one does not wear clothes made from animals.
There is probably no problem with a Jewish woman being completely vegan in this sense (though there's the issue of owning mezuzot). But men are required to use tefillin (made from leather) and tallit (wool).
There are in fact "kosher vegans" who eat vegan but use kosher tefillin. Rabbi Shmuley Yanklowitz has started an organization to support them (called Shamayim V'Aretz.) The head of Jewish Vegetarians of North America, Richard Schwartz, is an Orthodox Jew (who is at least vegetarian).
The ideological aspect is potentially problematic. Many vegans believe it is inherently wrong to ever use animals for food. So if this is the motivation for being vegan, it contradicts Judaism, which permits certain animal foods. For this reason, some say it is forbidden to be a vegetarian or vegan for "animal rights" reasons. However, because factory farm conditions cause unnecessary suffering to animals, and unnecessary suffering is forbidden (the mitzvah of ba'alei tzaar chaim), one could be vegan to avoid benefiting from or supporting probable violations of this mitzvah.
Another consideration is that veganism is an all-encompassing lifestyle and ideology for many people. A possible danger is that Jewish vegans will find this lifestyle and community satisfying and no longer feel the need to believe in or observe Judaism. This shouldn't prevent Jews from becoming vegan for reasons of health or ba'alei tzaar chaim, but they should avoid getting too invested in vegan ideology, community, etc.
Even if you are actually completely vegan, meaning not only not eating meat or any other product derived from animals (such as milk), but also checking for bugs and not using utensils that were used for meat or other non-kosher things, it wouldn't be enough.
Keep in mind that terefot are not the only halacha in cashrut. For example, there's also nesech wine, orla and bishul oved cohavim.
There's also the question of whether abstaining from meat is permissible.
First of all, there's no obligation to domesticate animals like you say, but if you do want to eat meat you need to slaughter. While there's no specific commandment to eat meat, there is a commandment to be happy on iom tov and joyful on shabat. It is explained that this is achieved by eating meat meals. Because of this, people make a point on eating meat on holidays, shabat and at other festive meals such as berit mila or weddings. On the other hand, you could argue that someone may dislike meat and hate its taste, so eating meat on shabat not only wouldn't achieve the proper goal but would in fact have the opposite effect.
Most religious Jews do eat meat, but I have heard that there are some that don't. I met one religious Jew who was actually vegan (we ordered pizza and he ordered one without the cheese) and one rav who had a certain revulsion to meat after a visit to a chicken slaughter house, so he would eat only a little meat on shabat just to fulfil the halacha.
I have heard some answers from rabanim from people dwelling with this apparent stira (being vegan). The conclusion was that while you shouldn't look to follow this path lechatehila or convince people to do it, there are those unique people that are vegan (or vegetarian) and I didn't see them trying to convince them otherwise.
Also, on a general note, the individual's health should always come first. It is common for vegans to lack some kind of vitamin and needed supplements, or they put their health at risk because of the lack of something in the blood.
In addition to Alex's thorough answer, it's worth noting that even a vegan diet might pose a problem of bugs, if fruits and vegetables are not washed / checked properly, based on the situation in your location (different vegetables require checking in different places, as the bug infestations differ by locale).
Eating out, even at a vegan restaurant, poses this problem, as well as those related to bishul akum (food cooked by a Gentile) and the general issue of neemanus (trustworthiness). [also see this interesting link about vegan products containing animal derivatives: http://www.quarrygirl.com/2009/06/28/undercover-investigation-of-la-area-vegan-restaurants/ ]
There's no specific mitzvah to eat meat (except in connection with certain sacrifices, but those are in abeyance until the Holy Temple is rebuilt - may it be soon).
There is a mitzvah to enjoy Shabbat and holidays, and for most people that includes eating meat. But it's somewhat subjective; if for a particular individual that would be a burden rather than a pleasure, then by all means they should eat something else.
The real issue, perhaps, is more one of outlook. If a person chooses to be a vegetarian/vegan for health reasons, for example, then there's nothing wrong with that. On the other hand, if their rationale is that humans have no right to use animals for food or the like, then we need to ask: why are they trying to be more merciful than G-d?