postcode or postal code?

This is less a grammar question and more a question of which terminology is used by the postal system in a particular country - for instance, in the UK this is a Post Code, in the US a Zip Code, in Canada a Postal Code, and so on. If you are developing for a specific country it would be best to use whichever term is used in that country. Wikipedia has a list of some country-specific synonyms, though I imagine it is not exhaustive:

CAP: Italy

CEP: Brazil

Eircode: Ireland

NPA: French-speaking Switzerland and Italian-speaking Switzerland

PIN: India

PLZ: Germany, Austria, German-speaking Switzerland and Liechtenstein

Postal code: Canada

Postcode: Many English-speaking countries and the Netherlands

PSČ: Slovakia and Czech Republic

ZIP code: United States and the Philippines

If you are developing for multiple countries at once you will have to either pick a term that makes sense for you or design a way that users see whichever is appropriate for where they are based.

To directly answer your question: When deciding on your database schema, you could use either postcode or postal_code.

An English speaker would be just as likely to understand either phrase, especially when looking at the context and the contents of the field.

That being said, although Wikipedia is not an authoritative source, the community on Wikipedia seems to have settled on Postal Code.