"I do not want paprikás, thank you."
Translation:Nem kérek paprikást, köszönöm.
"Akarok kérni" is too much. "Kérek" is enough to say you would like to have something.
If you want to be very polite, say "szeretnék kérni" (lit. "I would like to ask for"), but that's not used in negative sentences like this one ("nem szeretnék kérni paprikást" is just weird). It also doesn't match the less polite English sentence.