So I know for an observer outside the event horizon, objects become infinitely time dilated at the event horizon, making them look as if time stopped for them. If I then throw an electron into a black hole with known momentum, and it "freezes" at the event horizon, don't I now know both its momentum and position, violating the Uncertainty Principle?
In GR the location of an event horizon is precisely defined, but GR is a classical theory so has no uncertainty.
We have no theory of quantum gravity, so we don't know what an event horizon looks like when we include quantum effects. There are no end of suggestions, e.g. the firewall theory, but the truth is that we simply don't know. However it is generally accepted that when quantum effects are included the event horizon will become fuzzy in some sense so its position is subject to some uncertainty.