Although I’m unable to comment on frequency, heteronymity can occur in Japanese. Some examples:
The word 風車 can be pronounced as fūsha or as kazaguruma. The first almost always means “windmill” and the latter is often “pinwheel.” The latter can also refer to Clematis patens, a vine whose flowers resemble pinwheels.
A more extreme example is 無心, which has only one pronunciation (mushin) but several definitions: “innocence” (as a noun); “insentient” (as an adjective); “to pester someone for money” (as a suru verb); and, in a Buddhist context, “free from obstructive thoughts” (again as an adjective). Maybe this isn’t a heteronym, though, since there is only one pronunciation.
The word 半身 can read as hanmi or as hanshin. The former refers, in sumo, to a “defensive position with one leg bent in front and other extended behind.” The latter means simply “half the body” or “half length.”
And then the word 縁 is just annoyingly multifarious. Pronounced as e, eni, or enishi, it means “fate” or “destiny.” Pronounced as fuchi, it means “the surrounding edge of something.” Pronounced as en, it means “fate” or “destiny” yet again (how unnecessary!), but also carries a Buddhism-specific usage and may refer to a “narrow open-air veranda.” Pronounced as heri, it means “edge,” “tip,” or “margin.” Pronounced as yukari, it means “affinity” or “connection.” Pronounced as yosuka or yosuga, it means “something to rely on,” “aid,” “clue,” or “memento.”
There are probably many more. I’d bet single-character words (like 縁 above) are especially guilty of this (as 分 is, which I just realized). After doing some research it also seems there are lot of heteronyms with special Buddhist readings.
Of course, the concept of “spelling” is bit different (maybe nonexistent) in Japanese, being that it uses both a syllabary and Chinese characters. I think someone mentioned as much earlier…
Very fun question.