IMHO, dispatchers are not legally required for this type of operation. They just have whomever they are getting to fly the planes push the buttons on the Dell. Or management, or admins. It ain't rocket science to punch in city pairs, times, registration numbers and get a flight plan. Then have the pilots interpret the WX and make adjustments as necessary.
IF they had dispatchers and IF those dispatchers were IBT, I doubt they would be crossing any picket lines.
AJT is 121 Supplemental ... doesn't require a dispatcher (lisenced or not) to issue the release. The flight release is issued by anyone that the GOM/FOM specifies. A flight follower is the only requirement and the flight follower can issue the release if the GOM/FOM so allows.
OTOH, most of these supplemental carriers do hire people with their dispatch certificates as it significantly reduces their insurance premiums. I am not aware of any supplemental carriers which are unionized, though.