During today's Wrestling Observer Radio, Dave Melter revealed that All Elite Wrestling doesn't currently have its own dedicated Wellness Policy, similar to WWE. While there is something written in the contracts of AEW stars, there's no policy in place for suspensions and the like.
The promotion does have the right to drug test wrestlers, however, but there's nothing to say any failures would have to be announced publicly (or that any of the company's stars will be suspended).
This is a little troubling in some respects, but AEW is a privately owned company, so they don't technically have anyone to answer to. WWE, on the other hand, has to report to shareholders. There have no rumours or reports of failed drug tests, so hopefully this isn't an issue.
We'll keep you guys updated if we learn more.