First thing I never heard of rain causing a toilet not to flush. You will have to explain more , are you on a sewer, are a septic tank are a lagoon. Maybe check out your toilet as, is water up to mark in tank, is your flapper holding, is the 1/4″ tube in the 1″ overflow pipe, are the holes in the rim of the toilet clesan, next time it wont flush take a 5 Gal. bucket of water and dump in bowl and see if it goes down. When it rains and bowl wont flush does it overflow, give me more info and let me know.
Art retired plbg