It's definitely plausible, because both of the things you're asking for already exist. All you need is a prey animal who evolved limb or skin loss as an emergency escape mechanism when attacked by predators.
For the biological tourniquet: vasoconstriction. Vasoconstriction is the narrowing of blood vessels through muscle constrictions, and is an important processes for stopping blood loss (hemostasis).
For why having this tourniquet is useful: autotomy. Autotomy is a self-amputation behavior designed to escape or distract predators by sacrificing part of an animal's body. Several species of lizards are able to detach their tails when stressed, and in some cases the tail continues to wriggle to distract a predator (a few can even regrow their tails after detaching them). One mammal, the spiny mouse, has autotomy, and is able to discard and regrow chunks of skin to wriggle out of predator's grasps.
Regrowing an arm or a leg is a bit more unlikely to develop than regrowing skin or a tail, mostly because animals who can (slowly) regrow limbs wouldn't survive long enough without them to pass on their genes. However, it's definitely possible, and if such autotomy is common in a species then strong vasoconstriction would also be a very useful trait along with it.