Can Service Workers respond to synchronous XHR requests?

In Theory

Yes, service workers should be able to respond to synchronous XHR requests. This isn't explicitly stated in the specifications, but there's no exception that would cause synchronous XHR requests to be treated differently, and the W3C Web Platform Tests (WPT) suite has a test case to verify that it's supported: wpt/service-workers/service-worker/fetch-request-xhr-sync.https.html.

In Practice

As of January 2019, service workers can respond to synchronous XHR requests in Firefox and Edge, but not in Chrome or Safari. Chrome is planning to add support soon, but we don't know if Safari ever will.

An up-to-date browser support matrix is available at WPT.fyi. You can run the WPT test case in your own browser at https://w3c-test.org/service-workers/service-worker/fetch-request-xhr-sync.https.html.

According to XMLHttpRequest spec, the fetch procedure for both synchronous and asynchronous requests is the same so, in theory, it should be intercepted but Synchronous XHR has been deprecated so I would not expect Chrome to fix this.