Balance it on your head.

How about getting one of those thermo-cups with a lid?

Hmm not a bad idea there... I think we have one lying around somewhere.

The lid was going to be my suggestion too. Maybe drink it from a jar with a screw on lid or something. That or ring someone & tell them to come over & bring you coffee

Right gonna steal the thermos mug.

Easy.
Drag all of the coffee making equipment to your computer.
Then sit down have a short rest
Make coffee.
presto.
Or as is already mentioned simply use a thermos

Thermos again for a suggestion.
I'm sure someone has made a little device that attaches itself to one of the crutches and allows a Thermos to be clipped onto it. So you don't even have to carry it. If they have not, then by golly, go out and make one and sell it.

Ooh! ooh!
I got one.
These crutches of yours, are they made of these hollow metal tubes with a rubber stop at the end?
I think you can figure the rest out yourself.