Topic on Extension talk:GoogleSiteSearch

Yes, it'll still work. You can still create CSEs through https://cse.google.com/ and use the CSE IDs.

Does this extension still work now that Google has retired the CSE API?

https://productforums.google.com/forum/#!topic/customsearch/HQisP6ckZbg

Yes, it'll still work. You can still create CSEs through https://cse.google.com/ and use the CSE IDs. The GoogleSiteSearch extension just embeds a Google iframe-type setup for that CSE, which is still supported normally.