I think everyone is overthinking the issue at hand here.
I figure they would be very shallow. Just enough to stop the feet from moving but enough to have a gap between the glass and the table. 1/8th to 1/4 inches deep at the most and less than an inch in diameter.
Really, all the OP is asking is how to make a circular recess 1/8" to 1/4" in depth, and less than 1" in diameter.
The OP mentions that he does not currently have a set of Forstner bits, nor a drill press. In my opinion, this would be the best tool for the job. For a hole 1/4" maximum, drilling freehand with a Forstner bit should be no problem whatsoever. I have personally drilled > 2" deep holes with a 2" diameter Forstner bit, freehand, with no ill effects to my health. Sure, the holes were a bit wonky, but that's not the point.
All discussion of making a template and using a router is way overkill. By the time one makes a router template, sets up the router to the correct depth, and routs out one of the recesses, I would have drilled all four holes freehand with a Forstner and moved on to other things. Considering the time invested in doing the router work, I think it's cheaper (and more efficient) to get a Forstner bit. Then, the OP will have a new tool for his arsenal that can be used down the road. (Isn't buying new tools half the point of woodworking anyway?)
Otherwise, a spade bit with spurs on the edges will do a decent job at this task while still being pretty cheap.
If done correctly, one should end up with a circular recess that looks like this:
All it takes is a steady hand, and maybe a little patience and iteration to make sure the hole is the correct depth. Let us not forget that this is pallet furniture. I'm not trying to put down the OP's craftsmanship, but this isn't exactly a Greene & Greene restoration job. A little bit of tearout will not be a huge issue and can be sanded out of the softwood of which this pallet is made. The OP can always chalk it up to the
unique charm of the piece.
Other drill bits could be suitable for this job as well. Please see this related thread for further reading.
Update: After reading some comments, I've seen that the OP indicated he may be making multiples of this table. In this case, the use of a template may be warranted since he would be amortizing the time invested on the template over multiple pieces.