Some have access areas to light the pilot light and covers could be missing so yes to your answer. Many times I have seen the hardware or bottoms have rusted so much that everything on the bottom is missing.
Rodents can squeeze through very small openings. Any portion that is loose, rusting, or not fully sealed could be enough.
But who knows if the rodent that was seen actually entered through the furnace. It could have easily originally entered from elsewhere, and simply used the floor furnace as a hiding spot.
I recently did an inspection where there were rodent dropping in the attic and all over the garage, yet none in the crawlspace. Go figure.