PE compared to HVAC license?
This is an interesting question, and like many of these questions a simple yes or no isn't going cut it.

First, a PE license grants you the right to practice the design and approval of systems, but does not grant you the right to preform certain work. As @Haprer has pointed out the PE must apply for a EPA license and probably a contractors license. So you would have to have more than one license. In my state it does allow a different path to a license. For example, in my state the licensing jurisdiction allows people with a degree in electrical engineering to apply for a master electrical license and foregoes the four year apprenticeship and training. Even then it does not allow you to perform your own work without a contractors license. 

Second, would be why would they do it. If you are a practicing PE you are probably trying to move to a career different than that of a contractor. Then there is the matter of liability. As a practicing engineer you or someone has to pay for your professional liability insurance and as a contractor you would have to pay for a different type of liability insurance. Plus the other cost of licenses and professional requirements. It would be like a person becoming a licensed medical doctor and then getting a job as a paramedic. It really doesn't make sense, most would really like to see their careers go the other way.

Third, as far a purchasing, a wholesaler has an entirely different set of rules about who they sell to. They are in the business to sell large amounts of specific equipment and material to a select group of people, and they are usually regulated by the state to only sell to particular parties that are licensed and approved to used certain products. All this requires that purchasers are usually licensed and are approved before purchasing. It's usually on a credit line and invoiced for payment every thirty days, because it is not a cash and carry retail outlet. So there is approval of licenses, credit application and other legal documentation before being granted an account.

So in answer to your questions I give you the ambiguous "it depends", unlikely but not impossible.

Hope this helps