Will rain water cause the engine not to start?

It could be because of a number of things getting wet (spark plug, carburetor, air intake), if it is not pressing that you use it soon you can either just wait for it to dry but it might be better for the mower to dissasemble it to individually dry the spark plug, air filter, fuel filter, carburetor, crankcase and combustion chamber. If you still have the user manual it should be easy enough to do.