You'll want to test against stable API/ABI, and use positive and negative testcases. The problematic parts are mocking, simulation and how tests affect the codebase.
- The only stable APIs I've seen, if an API exists, are production financial APIs, because money is involved.
- DynDNS has been knows to change input and output parameter types, breaking calls
- Even Twilio has changed a fundamental API path (!) in the past couple of years, breaking SMS API calls in 2019
- Facebook only supports API version N and N-1. Hope you're not on N-2 and there's a sudden version bump!
(I say production APIs, since even payment gateways often have flaky dev/qa gateways that you can't reliably do automated tests against. Past companies that I worked at had to test against prod gateways with their own personal credit cards.)
Otherwise, if you want an API, ensure you choose a product/partner that offers/supports one.