"We are vegetarians."
'Vegetarianism' is pretty a pretty new concept in European countries, so it's no wonder the word for it is new. The concept first appeared in England, so the English word was borrowed to other languages.
Вегетаріа́нець is the word used in Ukraine. It was borrowed from English, but a suffix was added to make it similar to other Ukrainian words, like мексика́нець 'Mexican', америка́нець 'American', etc. It's also a bookish borrowing: it was borrowed by people who wrote English but didn't speak it often, so that's why it follows writing (vegetarian = вегетаріан-) and not pronunciation.
Вегетеря́н (or вегетаря́нин? do you add a suffix in singular?), on the other hand, was borrowed in its spoken form, so it follows the English pronunciation.
It actually makes sense: American Ukrainians spoke English so they borrowed spoken form. Ukrainians from Ukraine usually saw English words in writing, but didn't hear them in speech, so they borrowed the written form.
By the way, what's the singular form? «Він вегетеря́н» or «Він вегетеря́нин»? I'm just curious.
I just completed session 9 of level 3 and have become very frustrated. Of the twelve (12) questions seven (7) were vegetarian as part of the answer. For a demographic that is less than 10% of the population to garner over 58% of a work session annoying. How about putting some more time into naming some of the other items that may be on a menu in a restaurant