Unfortunately, there is no simple answer to this, and researching through blog posts, forum discussions and YouTube videos will inevitably result in lots of contradictory statements and claims. Below we will briefly discuss the science behind using vapor barriers, before making some recommendations (all of which come with important qualifiers).
Vapor moves from high humidity to low humidity, which means that it will move from a warm area to a cold area. An insulated garage is usually warmer than the outside and the vapor will therefore try to move through the wall materials and escape the garage. It will diffuse through the drywall, insulation and then the sheetrock. Some people refer to this as allowing the garage to breathe.
Now we will discuss some of the contradictory claims. Some argue that installing a vapor barrier will trap this vapor in the garage and result in damp and mold, and therefore it should not be installed, especially in the southern states which have warm climates and are therefore likely to see a lot of vapor movement from inside the garage to the outside. It is argued that it should just be allowed to travel out through the wall materials. The majority seem to agree that insulated, unheated garages, should be allowed to breathe.
If the garage is heated, you will need a vapor barrier, otherwise you will get frost and mold. Some argue that the car itself will cause the garage to be heated, but this would only be the case if you left the car to warm up in the morning, otherwise it would not have the chance to warm the garage. Others argue that a vapor barrier is required to meet building code. For this, I would check your local regulations as they vary from state to state.
We hope this helps, but if unsure, contact a reputable contractor for their advice.