There's two issues here IMO, the site name and the URL.
I agree with those who oppose using "vegetarianism" alone. It can be confusing when coming on the site with a vegan interest. It is not obvious to everyone that both fields are covered: "vegetarianism" is not a perfect superset of "veganism".The term "vegetarianism" is generally understood as focusing on a meat-free diet while "veganism" goes far beyond food, including everyday things like shoes or even money.
I personally don't mind a super long site name: in everyday use, it will find an accepted abbreviation right away. Software Engineering, English Language Learners, or Theoretical Physics aren't super catchy names either.
I'd vote for making the site name
Veganism & Vegetarianism
as intended. (Or the other way 'round, whatever.)
"V&V" would become the accepted abbreviation in a heartbeat.
The URL is a different thing: even just "vegetarianism" makes for an awfully long URL with the ".stackexchange.com" domain name. Remember, this is going to be the site's URL forever.
I'd suggest an abbreviation that should be acceptable to both vegans and vegetarians, is short, and fairly unambiguous: