.... which look up a user's address when they input their postcode (zip code)?
I can't imagine how you'd look up the address from the zip code as there could be millions of addresses within a zip code, but you should be able to write a script that can extrapolate the zip code from the address using the USPS Zip Code Lookup API [usps.com].
There are code samples in the docs to get you started in multiple programming languages.
UK postcodes apply to a small set of properties and can be used to offer a pick list of addresses. Typically it would be the number of addresses whose letters the postman could carry in his hand at one time.
You would have to pay for access to the file.
If you are dealing with multiple countries you would need to deal separately with each post office or find a third party processor who has done this for you.