You're going to have a hard time finding fully-waterproof sneaker-style shoes because the nature of a sneaker implies holes for ventilation and a slit between the tongue and the upper.

I'm aware these aren't what you asked for, but since that doesn't exist you might have to consider them anyway:

You could get some cycling shoe covers, although they're intended to go over roadie cycling shoes for use with clipless pedals (3-post cleats), not regular sneakers.

You could also get mountain-biking shoes (2-post cleats) or trail-running shoes, but they won't be fully waterproof.

Personally, when I was commuting a decent distance by bike (far enough to justify using clipless pedals), I wore cycling sandals when it was warm enough and mountain-biking shoes when it wasn't, and then changed into dry non-athletic shoes that I kept at the office.