The answer is that it can be done. However, here are the things a broker provides which you will need to get if you go solo:
1. The correct form of a timeshare sale contract that complies with Florida real estate and timeshare laws (if you don't do it right you will have a contract that most likely will cause problems).
2. A lawyer or law firm that carries out all the legal requirements, including likely having drafted that form contract and all the necessary statements for closing (you would need to retain a Florida lawyer with knowledge of timeshare real estate transactions).
3. An escrow bank account to hold any funds given as a down payment and through which funds are deposited and paid at closing (a real estate lawyer could also provide that).
4. A title insurance company that provides necessary title insurance, the place for the closing, and complete all acts necessary for closing, including getting funds issued from the escrow account to the seller and for taxes, and making all necessary filings with the recorder's office (you would need to retain a title insurance company in Florida; real estate lawyers can usually arrange that) .
5. A readily available market of buyers (you would have to market the property yourself unless you already have a buyer).