Could you also say 'Blommor gör mig att nysa', or is this a much more natural way to say it?
Ahh, I see. So a sneeze is something you 'receive' rather than are made to do, so to speak?
I am not sure that I understand what you mean. "Blommor" is the cause. Literally, "blommor gör att jag nyser" means flowers make that I sneeze, so it is very close to the orignal English sentence.
The expression "det gör mig" (it makes me) works with adjectives in Swedish but not with verbs:
Blommor gör mig glad - Flowers make me happy
Blommor får mig att hoppa och skrika - Flowers make me jump and shout
It "gives me a sneeze" is just about the cutest phrasing I have ever heard! Haha