SQLite.
the engine (sqlite.dll) is probably already installed on your machine, so you'd just need the ODBC-driver. Plenty of sources on the net.
Use DB-Browser for SQLite to design your Database, use whatever Language(IDE) you like to write code.
regarding multiple-client access. there is pro and con for SQLite.
As long as you don't have multiple clients trying to WRITE-access the same row, SQLite handles concurrency reasonably well.
So if your multiple clients access it at the same time just with Reading-Requests you should'nt have any problems.

BUT: to make something clear: A dedicated DB-Server is still the right way to go. If you'd like to do that via portforwarding (which still means exposing it directly), or via webservice, is up for debate.
There are a lot of Web-Space-providers offering "ready-to-use"-packages incl. DB-servers (MySQL)