I did do a bunch of Google searches before posting here, but this is a surprisingly difficult thing to find. There are countless web services out there to provide city/state based on zip code, latitude/longitude based on zip code, distances between zip codes, etc. But it's extremely hard to find anything that provides zip code by address/city/state.

That said, I eventually found one that looks somewhat promising, although I haven't had a chance to test it yet:

http://www.dev1.eraserver.net/...zipcoderesolver.asmx

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