Some will and some won't. If they are hit so they don't die immediately they will have a physiologic induced thirst. As the blood volume drops the body starts to pull fluid into the vascular system to try to restore voulume and pressure, this fluid shift causes dehydration and hence thirst.

The deer may or may not act on the urge but it will be there. Some may be too sick/weak to get up and try to get to water. Some may think hiding and staying put are in their best interest.

Some go to water some don't. There is a physiological mechanism to send them there. They like most creatures with higher level brain function can overridel it.