Yes, a stamp is required. Every piece of mail going into a federal prison or any correctional facility must have postage affixed before it is sent. No facility accepts mail with postage due or pays for incoming mail on behalf of the sender. That is not how the postal system works for correctional facilities or for anyone else.
A standard first-class stamp covers a regular letter going anywhere in the United States, including FCC Coleman in Sumterville, Florida. If you are sending multiple pages or anything heavier than a standard letter, weigh it first to make sure one stamp is sufficient. Underpaid postage results in the letter being returned to you rather than delivered.
The address for Coleman Satellite Camp needs to include your mother's full legal name, her federal register number, and the complete facility address. Getting all of that right before you drop it in the mailbox ensures it reaches the right person without being delayed or returned by the mailroom.
If you want to avoid the hassle of printing, addressing, stamping, and mailing yourself, InmateAid handles all of that for you. You write the letter online, InmateAid prints it and mails it with proper postage through USPS, and it arrives at Coleman's mailroom within two to three business days. The return address on the envelope is InmateAid's address in Florida, which keeps your personal home address private if that matters to you.
Either way, a stamp is not optional. Make sure it is on there before anything goes out.