My husband bought me both a Spot Bot and a Rug Doctor. Yes, the big one like you rent at the local Walmart. I frequently clean my carpeted areas and they come out fine. BUT you do realize that urine and feces have numerous bacteria in them right? So be sure you get a bottle cleaner for the machine that disinfects too.
I used the bottles of cleaner I buy alongside the machines in the housewares area. They're right under $20 per bottle. No rinsing required. I've used many things and these just do not leave a residue.
Also, use the hottest water you can. It makes a huge difference in the ease of it coming up.
All a wet vac does is vacuum up fluid, it does not work well for cleaning.
Here's my opinion on the lower cost carpet shampooers.
The brushes/bristles are the only thing that matters. Period. They all suck up fluid. The bristles that go round in circles side by side, they leave a fine matted up mess on carpet after it's been cleaned. It mattes it up. It has brushes sided by side going opposite ways and that is just too hard on some carpets.
The Rug Doctor has one long paper towel roll shape and it spins around like a roll of toilet paper does. One long bristle/brush going one direction. It doesn't leave the carpet matted up.