I own several class III items so I hope what I have below helps:
First, be absolutely sure your STATE laws allow you to purchase a suppressor.
Next, if you want to purchase it in YOUR name you must get the signature of the chief law enforcement official in your area. This could be, and most likely is, the sheriff or it could be the chief judge for your courts, or it could be the local district attorney. But going this route has it's pitfalls. The biggest one is politics. If they don't know you very well, they probably will not sign for you.
The workaround is to purchase the suppressor in the name of a living trust. There are software programs that will allow you to setup a trust, but I would use an attorney to do that for you; but it costs much more to go that route. To be on the safe side, I would contact a local class III dealer in your area that has been doing this for some time and they can guide you through the process.
The hardest part of this is deciding on which route to go. For me I used my living trust, which I had created several years before I purchased my first sub machine gun. It was very straightforward and it took approximately 6 weeks for the transfer to go through. I understand now the wait time is around 90 days.
Good luck!