security - How should I approach asking to have a lock put on my storage cabinet under my desk? - The Workplace Stack Exchange
A cabinet of individual key lock boxes
What's your budget?
I haven't found anything I would consider "reasonably priced", but if that's not an issue I found a few things searching for "cluster boxes"
https://www.ebay.com/itm/364990058834
https://www.homedepot.com/b/Hardware-Mailboxes-Multifamily-Mailboxes/N-5yc1vZc8eu
Edit: Sorry, I just realized these need keys, not combinations. Well, the links might put you on the right track, anyways.
More on reddit.comRecommendation for adding locks to this cabinet?
Where can I get a small cabinet, with a lock, to store the OneConnect box?
What can Key Cabinets protect against?
Can Key Cabinets be returned?
What are the shipping options for Key Cabinets?
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.
I usually just put a lock on myself unasked, I've never been told to take it off. It's my gear, my keys. If no lock can be put on I don't store valuables in my desk, If I must have them at work, I'll have them in a bag which I keep with me.
You should never rely on anyone else to protect your valuable property. Some businesses actually ban employees from keeping personal valuables at work and explicitly inform them that they're responsible for their property.