Hi Anne W -- Natalia from the Pilates Anytime team here!
You can absolutely take this class with a regular yoga mat! There may be an exercise that involves holding onto the dowel that sticks out the sides of the High Mat, but you will still be able to perform the exercise by keeping your palms flat against the floor.
If you have any additional questions, please feel free to reach out to us at support@pilatesanytime.com