First, you would have to write the vendor a cease and desist letter to put the vendor on notice of the non-competition agreement. Without notice, the vendor would have a defense.
You really need to retain an attorney to evaluate the non-compet agreement. The Texas Covenant Not to Compete Act has specific requirements for a valid non-compete agreement. If the non-compete is not valid, you could be wasting a lot of time and effort.
You may wish to take advantage of my hour of free consultation for new clients. If you make an appointment, please bring copies of all your pertinent paperwork.
I agree with Attorney Erikson's analysis. I want to add that you may be looking at the wrong party. I think he's highlighted that, but I wanted to reinforce that you probably need to be issuing written notice to the employee, reminding him of breach of the agreement. Regardless of whether the noncompete is enforceable in a court of law, putting the employee on notice may open up the lines of communication, as you say you have not heard from the employee. You could send him the notice yourself, or hire an attorney with a limited scope of representation to draft the letter to the employee. If the employee is valuable enough to put under employment contract, he or she is probably valuable enough to have an attorney draft the letter. Did you have an attorney draft your employment agreement? If not, it may be time to have a Texas attorney revise your agreements so that you have certainty that your agreements will be enforceable in the future.