I would use the first, but "rainfall" is one word.
Rain fall will continue into the fall.
Rain fall will continue to the fall.
Are both okay here?
In BrE, that would be "Rainfall will continue into [the] autumn". Mind you, in the UK, rainfall is likely to continue into and through all four seasons!
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
I disagree slightly. "Into the fall" (autumn) would include some of autumn (perhaps only 1 day). If one wanted to include all of autumn, one could use "through autumn". The problem with subtleties in interpretation is they only work if the speaker and listener both agree on the meaning.
Raymott doesn't specify that it will continue throughout the fall, so, while I agree with your interpretation, I don't see that there is necessarily a distinction there.
Is it just me, or does this sound like it is raining now and it is not going to stop until fall arrives?
Maybe "rainy weather will continue..." or "Daily rain will continue..."