Well, it's not just the water tank, but also the pump and the water lines I'd be concerned about. Were the water lines, pump, and water heater all empty for both winters?
The tank may have survived - it would depend on how much water was in the tank, and how cold it got over the winter. Just because the tank didn't leak that you could see, doesn't mean that a crack didn't develop along the top of it. I'd start with filling the tank completely, and then checking for leaks.
Yes, that big piece of plywood under the trailer is what holds the water tank in place. You can remove the plywood to get access to the water tank, but be aware that on your 23 foot trailer, you'll probably need to remove the front stabilizer jacks in order to completely remove the plywood panel. Plus, those panels do not slide out very easily. If you do decide to remove it completely, the tank will drop out of the bottom of the trailer, so be sure to disconnect all the plumbing lines that attach to the tank. Fortunately, they are all located in the same access hole inside the trailer where you probably found the drain valve.
I would suggest that you slide the plywood plate forward a couple of inches, and then fill the tank with water. Then if you have any leaks, you'll be able to see the water running out of the opening you created by sliding the panel forward a bit. If all stays dry, then you should be ok as far as the tank is concerned.
Oh, and be sure to sanitize the tank and water system before you use it!