There are gyms in Tobago. I would check first with the hotel you are staying at, as a lot of them have in-house gyms. If it doesn't, just ask them at the front desk for the nearest gym or health club.
Most of the places have gym. In Tobago also you have a gyms. you can get the details of all the gyms available with the help of the guide or with the help of net you can find out the exact address and details of the gyms available in Tobago.