The clean way would be to establish a middle tier on the raspi, that is a webservice which can read from and write to the sqlite database.
Much faster to implement would be to install Cubesql server on the raspi.
If you operate it with raspbian, a debian based linux system, then you can handle it like any Ubuntu server.
The cubesql installation instructions for linux can be applied to the raspi as well.
Cubesql of sqlabs.com is the multiuser version of sqlite. In your code you mainly have to change the connection type and code, otherwise it is probably the same (dont know your code…)