Is being a vegan completely okay according to halacha?
Even if a vegan is not eating anything traif (non-kosher), perhaps it's not truly correct because you are not following the commandments to domesticate animals and eat meat.
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?
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.
My answer will address both vegan and vegetarianism.
The Bible is fairly clear that God's ideal diet for all of creation was vegetarianism. And that at some point He allowed the consumption of meat, under various caveats, and usually with disapproving tones. Some people erroneously try and say that mankind was given dominion over the animals, and therefore we are allowed to eat them, however, when that dominion was given, we weren't allowed to eat animals until many hundreds of years later. So whatever that "dominion" is, it has nothing to do with eating animals. Let's trace vegetarianism in the Bible, and then we will go to Rabbinic comments endorsing vegetarianism.
So in the beginning, God only sanctioned one diet for all life, that of vegetarianism. This holds true even to the time of Noach, for in the ark the Noach and his family. It's a verse that is often glossed over, but Noach and his family ate the same food as the rest of the animals, which we find from a later verse shows that Noah and every animal on the ark were still vegetarian.
How do we know they were still vegetarians? Because after the flood God says that Noach can now eat meat, just like the green of the field that was already commanded to be eaten. And there's a caveat, now that meat is permissible, animals are now going to be scared of humans. When all life was vegetarian, there was no need for the animals to be scared
It is worth noting that as soon as God decides not to flood the world again, He immediately launches into the permissibility to eat meat. This eating of meat allowance applies not only to humans, but also to the animals. And the placement of this allowance seems can seem out of place if one doesn't focus on it. As Rabbi Soloveitchik points out, it's possible that one of the reasons that God flooded the earth was because there were large segments of humans and animals who were becoming carnivorous. And therefore it could be that God sanctions meat eating in order that He doesn't have to destroy the world again.
From this point the topic of diet isn't discussed much. Eventually the Israelites are redeemed from Egypt, and there is further meat restriction. Noach and his family were allowed to eat meat from all animals (with caveats of fear, and not eating flesh from a living animal, etc), but the Israelites were to be limited to eating only vegetarian animals. And in instances in which the Israelites demand meat (rather than just crying out for hunger in general), God responds in a very unhappy way, and always describes the desire to eat meat with the Hebrew word תַּאֲוָה, which means to lust. In general, the word lusting can either be negative or positive depending on context, but with meat eating it's always negative. The first story happens in numbers. There's the initial lusting of the people, followed by a lot of dialogue, back and forth, and finally a plague due to the lust for meat.
Originally God was only going to sanction the eating of meat when one brings a sacrifice to the tabernacle. But God changes His mind due to the fact that the Israelites will lust after meat, and that since the tabernacle may be too far away, there is a risk that they might sin and slaughter the animal in their gates rather than make the pilgrimage to the tabernacle. So if one lives too far from the tabernacle, one may slaughter an animal to eat it in their town. However, it also implies that if you live close to the mishkan, you can't eat meat without sacrificing it at the altar in the mishkan.
To those that might view this "lusting" being neutral, one should contrast it to how God speaks of vegetarian foods. The distinction between how God views meat eating vs eating vegetarian becomes abundantly clear when contrasted with how God describes the latter.
Among many similar statements by the prophets are:
I am not aware of any instance where eating vegetarian is described with the word lust, nor am I aware of God ever waxing poetic with descriptions of meat eating like in the above passages.
As for Rabbinic views endorsing vegetarianism, here is a good summation, one can follow the link to the source at the bottom if they want access to the sources for themselves.
So is being vegan or vegetarian "kosher"? The answer is most definitely yes. However, this is not normative Jewish practice, and nearly every Jewish community has a practice of regularly eating meat. And the vast majority of poskim encourage meat eating, especially when it comes to holy days like Yom Tovim and Shabbath. However, there is a lot of misconceptions that people have about meat eating in the bible. There are those who think that eating meat is commanded, or even encouraged. When in reality, the truth is closer to the Bible preferring vegetarianism, and begrudgingly allowing the eating of meat. And noted modern Rabbis who are vegetarian for these (and other) reasons even include the former chief Rabbi of England, Sir Rabbi Jonathan Sacks.