The easiest way to handle this would be to bring your own lock. Depending on the cabinet's configuration that might not be practical.
Anyway... You asked:
I keep prescription medication in my drawers. Does this affect my case?
This makes your case 'open and shut' in my opinion.
Is it OK to ask for a lock to be put on the drawers? How can I approach the matter?
"I have prescription medications that need to be secured in my cabinet.
How can I get a lock for it? It can't be a shared key."
Asking for no reason whatsoever might raise some questions.
Asking to lock up medications is completely different... it is always acceptable.
