It doesn't work, unfortunately. Current version of Alby improperly depends on localdb storage in the browser, which can't (shouldn't) be enabled in the Tor Browser because it opens vulnerabilities. I actually wouldn't be Alby because of this, but I found a (terrible, painful, not recommended) workaround that let me limp along.

Many nostr webapps fail entirely in the Tor Browser due to missing exception handling for when localdb writes fail -- like snort.social -- and others initially appear to load but then don't function when you click buttons that write to localdb. Credit to snort.social, because I think they're actually looking at doing a patch to fix it, which is nice because I'd like to try it. Other webapps do work just fine (e.g. https://iris.to, which I'm using right now in the Tor Browser), because either they've written in fallbacks for when localdb is unavailable, or are written to avoid the problem, or... I don't actually know, but somehow they work.

For browser extensions like Alby, nos2x (on github) is an open source example that works fine. Please look at how they implemented it and push an Alby update. Because anybody privacy-conscious enough to use Nostr is probably privacy-conscious enough to be tempted by the Tor Browser, and in the long term, if people like me have to choose between Alby and Tor, Tor comes first.

Best regards.