This is funny and sad to me. We had SQLite in the browser[0]. I only did a little bit of work with it but it seemed actually pretty nice. It was torpedoed because it was SQL-based (and not trendy "key value" and "web scale"). There was the whole excuse that the specification was "whatever SQLite does" and, therefore, not suitable for being a standard. There would be worse things than SQLite upon which to base a stand…
Otherwise you'd have to deal with different versions of SQLite in different browsers, most likely outdated, with many options turned off. SQLite is full of quirks and gotchas, so it's safest to ship your version in your app.