Here's the deal. You land in Orlando, you skip baggage claim, you go to the DME counter and then to your DME bus. If you have carry-ons, you can have the driver stow them in the cargo bay of the bus, and you'll get them right back as soon as you arrive at your resort. In such an instance, I would definitely tip the driver the customary $1 - $2 per bag. But if you have no carry-ons for the driver to handle, and the driver didn't provide you with any personalized service, then a tip would be nice but it's not at all necessary.
Now, back to your checked bags, your suitcases, the ones that DME will deliver to your room for you. The Bell Services cast members who deliver these bags to your room are, believe it or not, pre-tipped by Disney, as part of the DME service. You should not tip them, because they are receiving a tip already. It is an actual, real extra payment they receive from their employers, and from what I've read the CMs compete over who gets to deliver DME luggage, either because it's bigger than guest tips or because it's a guarantee of a tip and they sometimes get stiffed by live customers.
Wow, are you sure you saw a tip jar on a Disney Transport bus??? Really? Wow. They'd be terminated on the spot for an offense like that. They are definitely NOT permitted to do anything like that.