Quick answer
Я вегетариа́нец.
ya vegetariánets — вегетариа́нец for a man, вегетариа́нка for a woman — plus “without meat”
The options
Я вегетариа́нец.I’m vegetarian. — male
Я вегетариа́нка.I’m vegetarian. — female
Я не ем мя́со.I don’t eat meat.
Что-нибу́дь без мя́са, пожа́луйста.Something without meat, please.
Здесь есть мя́со и́ли ры́ба?Is there meat or fish in this?
без plus genitive
“Without” is без + genitive: без мя́са, без ры́бы, без лу́ка, без майоне́за. This is the most reliable phrase of all, because it names exactly what must not be there.
Hidden meat
Soups are often made with meat stock, and сала́т frequently contains sausage or chicken. Two useful checks: На мясно́м бульо́не? (is it on meat stock?) and А в сала́те есть колбаса́?.
What you hear back
Есть о́вощи на гарни́р.There are vegetables as a side.
Суп на мясно́м бульо́не.The soup is made with meat stock.
Мо́жно без ку́рицы.We can leave the chicken out.
Возьми́те варе́ники.Take the vareniki.
Typical mistakes
Я вегетариа́нец — said by a woman
Я вегетариа́нка.
The word has a feminine form.
без мя́со
без мя́са
без always takes the genitive.
Check yourself
Which is correct?
Genitive after без.