I'm not quite sure if you mean 'can i use sqlite (websql) in chrome' or 'can i use sqlite (websql) in firefox', so I'll answer both:
Note that WebSQL is not a full-access pipe into an .sqlite database. It's WebSQL. You will not be able to run some specific queries like VACUUM
It's awesome for Create / Read / Update / Delete though. I made a little library that helps with all the annoying nitty gritty like creating tables and querying and a provides a little ORM/ActiveRecord pattern with relations and all and a huge stack of examples that should get you started in no-time, you can check that here
Also, be aware that if you want to build a FireFox extension: Their extension format is about to change. Make sure you want to invest the time twice.
While the WebSQL spec has been deprecated for years, even now in 2017 still does not look like it will be be removed from Chrome for the foreseeable time. They are tracking usage statistics and there are still a large number of chrome extensions and websites out there in the real world implementing the spec.