Some training classes do allow the puppies to play after the training, though often I find this turns out not be a great idea. By the time the class is finished and puppies are allowed to play, they are already a bit tired which could lead to "crankiness", also during training treats are involved this could also affect the situation with some puppies looking for more treats, and finally many such classes are held in local schools/clubs where the floor is wooden and too slippery for puppies to run and tumble while playing.
When I had a puppy and no suitable playing companion, I just took him for a walk in a local park in the hope to meet a play mate. And we did - not every single day, but often enough for puppy to learn and enjoy the close interaction with other dogs. What helps is that it doesn't have to be another puppy for him to play with, it could be an adult dog just of similar size, and of course friendly.