Hi Mike. Thanks for the picture, but that still doesn't give us a lot to work with. I assume you're trying to anchor him so when he's activated he doesn't vibrate across the floor? Is it static or have movement? Or you could want to anchor him down so he doesn't get stolen? We need some more info. If the chair was secured, would that help? If the shoes were secured, would help? Do you have access to the underside of the floor? Come back.
Do you have access to a wall on the back? Two little 1/8" dia. concrete anchor bolts in the floor on each side of the chair wouldn't be noticeable the rest of the year if it's your property, as with holes in the wall. You're Mom will never find out, haha.
Something fast and not too complicated would be to get some good heavy barbells or weights, then tie, tape or hook them so they are secured to the bottom supports of the chair. Then take some cardboard, paint it black, and cut it down, and tape it together to form a box OVER the weights to hide them. Make the cardboard cover the same width of the bottom of the chair area - so that it just looks like a platform under the chair itself.
You could also use fabric or black plastic to create a curtain type of enclosure (just tape to the underside of the chair, then tape down to the bottom part) to hide the weights, but you might need to position the weights across a beam of wood that sits cross-ways from either side of the chair's bottom braces...