What would happen if gray wolves became extinct?

As with the extinction of any other species, loss of the gray wolf could have a significant effect on the food chain(s) and ecosystems of which it is a member. Think of how a drop of water creates ripples on the surface of a lake—the loss of the gray wolf would at first have immediate impacts on its natural predators and prey, and eventually disrupt the food chains these other organisms are a part of.

Canis lupus, the gray (also spelled "grey") wolf, is found naturally throughout North America. Gray wolves' diets typically consist of large animals like deer, elk, or bison, but if these large animals aren't available, these wolves will eat rabbits or other small mammals. If the gray wolf were to go extinct, the populations of their prey—the deer, rabbits, and so on—may increase due to a lack of at least one natural predator. Especially in food chains where there is an exclusive predator-prey relationship, the population of the prey can essentially grow out of control with the loss of a predator. This can have secondary effects which deplete the food of the prey or cause a boom in the populations of other predators. 

Gray wolves are sometimes prey to larger mammals like bears or mountain lions. The loss of the gray wolf in an area where bears or lions rely on the wolf population for a source of food would significantly disrupt the food chain.