I am currently trying to access a SQLite database via an ODBC connection but the system that I am working on is extremely locked down and it is therefore not possible to install the drivers. Is there any way to bypass this?
The answer will depend upon what tools you have available. What database system are you trying to get the data into? Oracle? Sql server? Windows machine or unix box?
One alternative could be install software on a 3rd machine. (your PC?) Extract the data and convert it into a suitable format to import.
Another option could be to export the data to a .csv file and then import that.
You could use Excel, MS Access or even sql express.
However, this is going to become part of a regular process. I suggest you try again to have the required drivers installed. That would be the cleanest and most straight forward option. Everything extra step you have to take costs time and introduces risk.