One word: rust. I have cats and this has been a recurring problem. Two vents have rusted out (house is now around 25 years old). If your house is typical, the round ductwork fits into a metal box leading to the floor vent. The duct comes into the box above the bottom and perpendicular to the box. When urine gets in the vent, it flows to the bottom of the box and rusts it out. The insulation surrounding the box becomes soaked with urine. The other problem is urination next to the vent, which will run down the outside of the sheet metal into the insulation and may soak into the subfloor and carpet padding or wood flooring.
Get a bright light and a stick and/or a long screwdriver, remove the louvered vent cover, and start poking around until you find the damage. You may have multiple problems. Usually replacing the vent box helps a lot. Subfloor and carpet padding may have to be treated or replaced.