If you can answer yes to the following:
1. ability to upload the source to the site,
2. ability to create a DB, DBuser, and DBpassword
3. Webserver has MySQL and a current version of PHP installed
then yes you should be able to run OSTicket.
Some things that you may also need to make your life easier:
1. ability to schedule tasks (cron or Task Scheduler)
2. ability to edit your php.ini to change mail config