I started withe a 4" fan and a plastic tub that fit over the register.
Find a fan that you can take apart or it'll be more work if you have to drill and pop-rivit it back together.
I had to drill rivits and remove the stand (watch your fingers...)
I attached the cover to the inside of the tub and carefully cut out the inside, leaving a little extra around the screw holes.
After re-assembling the fan, I drilled a hole for the cord.
The finished product with cool feet.