Google’s John Mueller said on Twitter that right now there are no webhooks or APIs for tracking issues reported within Google Search Console. So what he recommends to do now is set up an email box that you can program a script to read those emails and feed them to your internal systems.
This is not as clean as using an API and yes, Google does change the text and content used in those emails over time, which can cause some issues for your scripts, but it is the only way to automate this process.
The Google Search Console API does not yet support this data but when it does, I am sure it will make this process a lot easier.
Here is what John said:
We don’t support that at the moment; the workaround we usually suggest is to set up an email address that you can poll automatically for new messages, and add that as an owner in Search Console.
— 🍌 John 🍌 (@JohnMu) March 26, 2020
Forum discussion at Twitter.