How to fix the damaged floor of a vanity cabinet?

I don't see evidence of animals. Obviously that area was very wet for a period of time at some point, and particle board doesn't tolerate that. A little pressure is all it would've taken to punch through like we see here.

The simplest way is to fit a new sheet of material over the top of the existing floor. If you use 3/4" material you probably don't need to do any repair to what's there now. Just cut it to fit inside the cabinet and extending out to the edge of the face frame. (Notch around the face frame center stile.) You may need to temporarily remove the drain trap parts to make room to finagle the new sheet in there, or even pull the countertop (with the faucet attached but disconnected from the lines).

Ideally you'll purchase a piece of shelving with a finished edge. Use that edge at the front. MDF isn't appropriate unless it's coated or varnished. It'll be a sponge for any spills or leaks.

If you don't like the idea of raising the floor that little bit you'll need to put on your cabinet maker hat. Cut away the existing floor, install small cleats all the way around, recessed the thickness of your new floor, and set in a new floor panel. That procedure is a bit much for an answer on a Q&A site, so post new specific questions if you like.

Whenever I replace or repair damaged sink or vanity bottoms it's usually with a material that will endure getting wet. If the original bottom is sound and firm I found laying a piece of 1/4 inch Hardiback or 1/2 inch cement board on top secured with screws works well. Once the new piece is in place it can either be tiled over (using thinset and ceramic tile and grout) or covered with vinyl sheet material. Painting is another option.

If the floor is beyond repair remove it by cutting it in sections. You will need to install some support around the perimeter to attach the new floor to. Rather than MDF or particle board, which deforms and weakens when wet, plywood or even 2x framing stock would last longer and be more durable.

Go to your local "restore" or used home materials store and just replace it. Either that or buy a new one. New ones will run you around $100 if you don't need faucet/sink. Used ones in good condition are usually $30ish. You will spend that amount trying to redo the bottom and doors.