What would you call a person who farts a lot?

Further to Thusagen's answer, the single word would be flatulent.

This person, in medicine, would have been called, "someone suffering from excessive flatulence"

I personally call such a person a fart-machine. ;)

But the correct term is flatulent.

I'd call them a fart-arse, a term which has expanded to include a number of other meanings:-

fart-arse (plural fart-arses)

(UK, slang, derogatory) A generic term of insult.

Verb

fart-arse (third-person singular simple present fart-arses, present participle fart-arsing, simple past and past participle fart-arsed)

(UK, slang) To waste time or opportunities; to dawdle.