In NYC the market is used to fees for everything. As it sounds like you manage the trainers who do the trainings, your fee would be for your services as the processing agent. You provide the service of getting the client into your system, taking their information, connecting them with the trainer, etc.
I don’t find it unreasonable for you to charge a registration fee. Most trainers don’t charge fees, but they don’t usually manage much more than their own schedule.