If I live in one state and wants to do business with someone in another state, do I need a notary public to make contract binding?

Do not write the contract yourself without a lawyer doing it for you. In general, whatever the two parties decide is a binding signature is one. So part of your contract is going to state that you both agree that whatever form you agree on for the contract will make it binding and admissible as evidence. The purpose of a notary is that the notary checks the person's identification to be sure it is actually the person signing. That can then be scanned and sent back as a pdf. The signature then also contains the notary's signature and license number. This is one level of being sure you are dealing with a real thing. The current list of California-licensed notaries is available online. If you have any doubts if you are dealing with the actual person, having a notary is nice. If you have any doubts, deal with the person's lawyer. And have your lawyer handle the transaction.

What you're describing is a simple licensing agreement. There is no need for a notary. As long as the agreement permits it, you can have the signatures signed and exchanged by email or fax.