Would the employee file stay on record?
Is it a risk of them finding out or not really?
If they don't have a central HR database then you could certainly get hired by another branch within the same corporation.
If they find out later then it would depend on company policy if you get fired again outright or if they take into account your employee evaluations since starting at the new branch and if the reason you were let go at the first branch was still an issue.
I'm guessing that they probably have at least some kind of HR or at least record keeping, but there is still nothing to keep you from applying.
Most companies have an HR department, even if it’s not a formal one. There is someone in charge to make sure that everyone got paid, that oversees the new hire paperwork and compliance with government regulations, etc.
If you were fired my one location, your information is somewhere.. whether it’s in the payroll records, timeclock, etc. Depending on the grounds for your termination, you may not be eligible for hiring at the other location. You can certainly apply and see what happens. Good luck!