I agree with AdC. A no-dogs policy may be the only way to go. You said you have cameras in your elevators. Can you adjust them so the area closer to the floor where dogs are can be viewed? Or reposition them so you get an aerial view of the elevator that will include people and dogs. Also, what will you do if/when you find out whose dogs are peeing in the elevator? If you don't have a house rule with a fine in place for this, you should enact one.

Too bad you have to replace carpeting if that's what you prefer, but an elevator isn't the best place for it - too much foot traffic, the nap flattens and it wears quickly. Keep in mind that a rubber/vinyl floor covering won't be good if you have dogs peeing in the elevators. It's porous, dog urine will permeate it, and the odor will linger. Not very good, especially in a confined area like an elevator. If you replace the flooring, go with something that isn't porous, or you'll be replacing it often because of the odor (and staining too).