making outgoing TCP connections (like which are needed for any networked database driver) is not implemented in WebPerl (I think).

Only partially: Emscripten supports WebSockets, and it exposes these via the C sockets API (see also), so actually Perl's sockets API is able to access it. On the server side, one would need a WebSockets proxy (e.g. websockify), assuming the server doesn't support sockets natively. This is as far as I got playing around with it on the WebPerl client side. However, the big problem is that these sockets are required to be non-blocking, and so the database driver would need to be implemented as such, which is a big ask.