EmBlaNce but they signed a contract which states that all the software is the creators property and we do not have access to it.
Well osTicket is likely not this persons property, so maybe talk to your company lawyers about this. It sounds to me like they could be in breach of contract but I am not a lawyer.
EmBlaNce i have access to the database via SSH, but not to the source code.
This means that you can dump the database to a sql back up file. That means that you can move your ticket system to another server. If you have SSH access then you should also be able to use FTP over SSH (SFTP) and download the sql dump.
Using this sql dump you can restore the Database to a new server. Setup a DB user account, and grant it privileges to the new databse. You can put the latest version of the product on that server and upgrade the database to it's format (when you log into the new server SCP the upgrader will run and upgrade the database for you).
Then you can look at the new system and compare it to the old system functionality wise. If this person who setup the ticket system for you made lots of modifications you may find that you lose some functionality. (Such as the aforementioned PBX portion.)
But at least then you would be able to make a comparison and decide if you want to continue moving forward with the transition, etc.