Might be able to do a little air, but liquid coolant isn't going to happen.
The machine just isn't set up for it, and most of the stuff we run on it would be damaged by any liquid.
Heck, the holding jig I made for the door tracks is made out of 1 1/4" thick particle board, it would turn to mush.
With routers we generally use chip load to pull the heat away from the part and bit. Big chips carry the heat away with them. Easy with larger tooling, small ones can get tricky. And counter-intuitively if you're bit is getting hot you increase the feed rate.
We do have a few Aerotech tool holders. They're basically a big turbine on the tool holder designed to pull dust away from the part and move a lot of air. All ours are set up for 1/2" shank tooling though, have to get some 1/4" collets for them.