Zillow has shut down all of their data APIs as of the end of February. As was stated in another answer, it appears as though they shut it down as a result of abuse, but did so with no mention of it at all (as a matter of fact, the documentation and sign up remain active to this day, nearly 3 weeks after the API began throwing 410 response codes).
In 2016 Zillow purchased Bridge Interactive, which is a company that specializes in Broker and MLS back-office solutions and data management. see press release here
Bridge Interactive now has the Zestimates API (see documentation here) along with a new API that appears to offer API access to public data too (Public Data API docs)
I had applied for access to the Bridge API in February using their web form on the Bridge Interactive Website shortly after the demise of the Zillow APIs and after about 2 weeks and a bit of back-and forth via email I just got my invite to access the Bridge API.
It does appear to have the same restrictions as the Zillow API just on their new platform. Sign-up is a manual process so it takes some time but, knock on wood, I might be back up and running in a short period of time after integrating with the new API (which, of course, is completely different than the Zillow APIs).
Update 2021-03-28
It appears as though on or about March 26th, their API began to respond again -- I have tested the getSearchResults and the Zestimate API and both are responding as they previously had.