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
I often wonder, in cases like this where the translation is of the meaning and not literal/word-for-word, what the usage is of each Swedish word. Since both får and att have more than one usage/translation, what's the literal translation of constructions like these? "Make me" isn't a word-for-word translation, but rather the translated meaning. So would the literal translation in this case be, "make me to", "get me to", or something else?