For a major rewrite I would choose FPC / Lazarus over Delphi, as it is more adaptable and the people on this forum give excellent help.
As for the choice of a database: as said, you don't want to maintain a database server on-site at each location. So, do your customers have internet access? If they do, use a big one in the cloud, like PostgreSQL. Make a database for each customer and make sure the users accounts can only access that one.
If they don't, it becomes tricky. As said, file-based databases on a share are a bad idea. Multiple users on an SQLite database isn't an easy thing to do. With an Embedded Firebird, you have to delegate one app to be the server. Tricky.
For security and safety, the big cloud platforms have all the tools you need: VPN connections, automatic management and backup. You can even deploy a different database for each customer. In the long run, that is the best bet.