This probably should have been posted in the NFA section.
Assuming suppressors are legal in your state (they are in most), you'll need to follow the BATFE's rules as per the 1934 National Firearms Act. There is no "permit" to own a suppressor; instead, each suppressor purchase requires the same process and the same $200 transfer tax to the BATFE.
The process is as follows: purchase the suppressor through your dealer, fill out some paperwork, take the paperwork and get photoed, fingerprinted, and signed off by local law enforcement, and mail the paperwork along with a $200 to the BATFE. In about six months your dealer will receive the paperwork back with a tax stamp affixed and they'll call you to come pick up your suppressor and your tax stamp. This tax stamp paperwork is what allows you to own the suppressor and it's a good idea to keep a copy of it with the suppressor at all times.
However, this process only works if your local law enforcement will sign off. If they won't (like in my neck of the woods), then you need to purchase the suppressor through a revocable trust. In many ways the trust is superior; it's a lot faster (the paperwork can be ready to mail out within about 15 minutes from start to finish), it doesn't require a sign-off or photos or fingerprints and it allows more than one person to be on it, meaning more than one person can possess the NFA item. Also, you can designate a person to receive your suppressors upon the event of your death. On my NFA trust I am the settlor, I am the first trustee, my wife is the second trustee, and my father-in-law will receive my suppressors in the event that both my wife and I die. The trust can be amended at any time, but I recommend you get advice from a lawyer or other qualified person to do that.
0331: "Accuracy by volume."