You can absolutely bring your own. We took in 2 on our trip last September. If you park in the parking lot just know that you have to collapse the strollers and lift them on into your seat with you (many times squished between your knees). But it is a short tram ride and everyone else is doing it too.
Yes, you are allowed to bring a stroller or 2 or 10 into the parks with you. Or, you can rent a Disney stroller. Or if you have a Disney Visa, you are given (allowed) 2 free stroller rentals per day.
The front of the trams have room for strollers that do not collapse. The front of all other tram cars have a wider area where one row faces backwards. There is a drop off area on Harbour just past the resort shuttle entrance but it is a pain to get out of as far as I am concerned.
Save money, bring your own stroller, and do not become a member of the pooped parent parade consisting of tired parents carrying dead weight exhausted kids to the car or hotel.