I am in Texas and don't know Michigan laws per se. However, Botox is a prescription medication and must have a physician of record regardless of state. Laws vary about just how directly involved in the actual treatment the physician must be. In many states, it is legal for a physician to delegate authority to a non-physician for such treatments but there are differences in the details such as if the physician need be physically on-site, have ever seen the patient, and similar conditions. Regardless of the state, if there is no delegating physician, then it is illegal. In addition, the delegating physician is ultimately responsible for the patient regardless of who does the actual injection.
Also, Nurse Practicioners (NP's) and Physician Assistants (PA's) have some prescribing abilities that also depend on state laws. These people have additional training and may have legal authority to prescribe medications.