I would say it's time for a new one. I wouldn't want to risk my dog or cat's health because of it being unsanitary. I would also place your concerns with the health department.

Public places no matter where should ALWAYS be clean especially in vet offices, animal shelters, hospiitals, resturaunts, etc. I can see it being gross because of the weather but if it's crusty now then they haven't been keeping up. Is it a big vet office?