Can I store the website screenshot URL, or does it expire?

You can store it. The screenshot_url on a website analysis is a Baselayer API address for that analysis, not a time-limited link to a storage bucket. Fetching it with a valid API key returns the homepage screenshot, and the same URL keeps working as long as the analysis exists.

Because it is an API address, the request needs your API key in the Authorization header like any other call. It cannot be embedded directly in a page for end users to load without going through your own backend.