The ticket system was re-installed. (Not removed)
I can no longer view old/current open, closed, and answered tickets. Is there a way to recover this data? It is EXTREMELY important to us.
I would verify that your original database is still present and contains tickets. Start there to make sure you actually have something to "recover".
Location
This was installed on a network solutions hosted web site. Where would the database be? Doe you know?
I would assume somewhere on Network Solutions then. Do you have an admin page for your website through them? Perhaps the SQL information is somewhere in there. For instance ours will show "MySQL 5 Databases", you then click on that and it lists the databases that have been setup.