Since we do not have all of the facts pertaining to your requirement, solicitation/contract or contractor(s), the following answer is based solely on the background and question provided. As we do not have access to the contract folder or particulars that apply to this situation, we highly recommend you consult the Contracting Officer and possibly the Legal Office.
In accordance with FAR 1.602-1(a),"Contracting officers have authority to enter into, administer, or terminate contracts and make related determinations and findings. Contracting officers may bind the Government only to the extent of the authority delegated to them." The answer to the question "Can a Commander authorize work to begin based on the position of Commander?" is no. Position or rank is not related to the authority to enter into, administer, or terminate contracts, to include ordering work to begin. Only a designated contracting officer, operating within his/her delegated authority, may authorize contracted work to begin. Continuing on, the question states that neither the Commander of the CMO nor the Contracts Director have contracting officer warrants.
FAR 1.602-3(a) defines Unauthorized Commitment as "an agreement that is not binding solely because the Government representative who made it lacked the authority to enter into that agreement on behalf of the Government." A Commander of a CMO or even the Contracts Director cannot direct work under a contract without the proper authority to do so, as described in FAR 1.602-1(a).
If the Commander or Contracts Director are not eligible for Contracting Officer appointments, a good solution would be to designate the two ACOs emergency essential, so that at least one of them is always available, at least by phone, 24/7. That way, the mission is not negatively impacted by the lack of their physical presence and you are not setting anyone up unnecessarily to create an unauthorized commitment.