If you're asking whether you need a lawyer to draft a contract, legally, the answer is no. Anyone can draft a contract on their own and as long as the elements above are included and both parties are legally competent and consent to the agreement, it is generally lawful.
A “submission agreement” (also called an “agreement to arbitrate”) is a written agreement between two parties that establishes the use of arbitration to settle a dispute (or any and all disputes) that may arise between them.