If your tarp is heavy duty enough, I'd expect it to work. Most blue plastic tarps are fairly thin and weak, so while it might work, it would be risky.
The difficulty is that you will be abrading the tarp by driving onto it, probably subjecting it to abrasion from rubbing against the jeep's body, and risking the tarp catching on underwater debris. The thinner your tarp, the higher your risk.
Even a pinprick leak could be a problem, depending on how long you are in the water. A flimsy tarp of any material will develop leaks or rip relatively easily, maybe even while loading for your first test. A sturdy one would probably hold up to a few trips, although I've got no idea how many. The toughest waterproof fabrics are used for whitewater rafting vessels, which gives an idea of how sturdy a tarp (not polypropylene) can be.
Make sure before you start that you don't have any leaks in the tarp. I've never had to do this, but a possible setup that comes to mind is hanging the tarp in a cup shape and dumping in 10 gallons of water or more, then swishing it all over the tarp surface to make sure it doesn't leak. At the minimum, I'd spray one side with a hose, but that has a few problems: 1) it's hard to be sure no spray is reaching the far side, which makes leaks harder to spot, and 2) if the material is only somewhat waterproof, it might still seem impermeable when sprayed, while holding several gallons would show that it can't stand up to long, heavy exposure.