Is it safe to drill into old (but still working) vacuum tubes?

The glass bulb of a vacuum tube (thermionic valve) rests on the base. In order not to damage the bulb, you should remove the base from the glass bulb first. You have to unglue the base and desolder the wires from the bulb. Otherwise drilling should be very careful not to break the glass.

Only large tubes provide sufficient space for installation of LED in the base.

Most tubes do not contain mercury. Mercury is only used in some (not all) rectifier tubes. It may deposite as shiny spots on the glass envelope (not to confuse with getter). The most widely known is the 83 tube used in many tube testers.

Vacuum Tube Light

I think you should call your manager so that you back yourself up. When my dad got forgetful, he had gotten into a car that looked just like the one he used to have, the actual owner left the keys in it and then, finally, when he got to a gas station and the gas station attendant asked he if he wanted a fill up my dad said he had no money. And then he said he wasn't sure why he was even there. Could be you have someone that has encroaching dymentia of some form. They called the police and found out my dad was missing and got him back where he belonged. You could have someone very deep in thought, or drugs or just doesn't want to be bothered. But the fact he couldn't find his room and he sounded lost point to someone in trouble. Short term memory is the first thing to go so these people often forget where they are going and are truly lost. I would call the manager(CYA) and be sure this person doesn't need help. You're a thoughtful person to knock on his door and double check and be sure he is ok. You can skirt around why you are actually worried but ask him if he needs any help and push him a bit if he hesitates. My dad was embarrassed as he got more forgetful. He knew he was forgetful and then would get angry when he couldn't remember but would be so thankful to be found after he wandered off. You need to do the right thing.

Vacuum tubes have vacuum in them i.e.: nothing. once you break the seal with a drill, they won't work anymore. It's fluorescent tubes that have mercury in them so that when the rarified gases in the tube become a plasma, the mercury vapour would emit UV light which is then absorbed by the phosphor coating and re-emitted as visible light. A vacuum tube will not have mercury because then the resulting ionized plasma would prevent it from functioning as a switch or valve. If it has mercury, it's not a vacuum tube.

Practice on incandescent light bulbs since they two have either a vacuum or a rarified noble gas for the very same reason, to prevent conductance by ionized gases. I doubt you'll be able to install LEDs and still have the tubes or lightbulb be functional.

neat idea - I used to work with some beautifully engineered beam tetrodes - the 807 and the humungous 813, and I thought about doing something similar - and mount it in a flower pot, with a fake Latin name tag like "Electrovolvus Thermionicus".
If it is that sort of tube (valve) then there is so little mercury as not to worry about - a tiny amount was used in what I think was called a Gettering pan (yup - just checked on wikipedia - see link).
If I was doing this, I would probably go in at the base of the tube, but cover it in a good layer of fibreglass body filler first, to keep the glass from shattering, then buy a drill bit designed for glass, and take it very gently. If you use 3mm LEDs, you could mount some inside a plastic drinking straw, to make it easier to insert them - I just bought some great multi-colour flashing LEDs off Ebay, so I will be making all sorts of crazy flashing things!