I am trying to gather a game plan of wanting to do all 4 parks in one day!! In my mind it all works out but I hit a roadbump... When leaving HS to get to AK will we have to go to a resort and then catch an AK bus from the resort or do they have buses that go park to park....? Then the same thing with AK to EPCOT. I managed to get the times worked out but now trying to figure in transportation.
We have done four parks in one day a few times, it was always the last day of a long trip during low crowd times. Our goal was to hit at least one or two headliners/favorites at each park. That was easier with paper FP, we could grab a FP as soon as we entered the park, then hop in a standby line for something else.
To answer your question, yes, there are buses between HS/AK/Epcot parks. The problem is that at 9:30am, there isn't much demand to hop between HS to AK, and the resort buses seem to run more frequently than park-to-park buses, so you could be waiting a looooong time for a bus. First time we started at AK at 8am EMH, rode Everest and Safari a bunch of times, then headed out to hop to HS. We were all alone at the bus stop, I mean totally alone, there was no one else leaving AK to go anywhere. Eventually a CM wandered over to ask us where we were going, he got on the radio and 10 minutes later we had a private bus ride to HS.
The next time we attempted four parks in one day, we started at HS, then to hop to AK we took a bus to AKL first and then AK. Another trick we used that time, hopping from AK to MK, a bus to the Contemporary came first and we just walked over.
And yes, there is a walkway between HS and Epcot, but it is not a short walk, it is probably at least a mile, something to think about if it the middle of the day and hot. The boat is an option but it makes a lot of stops and is slow. If I was attempting four parks in one day again, I would drive. It is remarkably quick to drive between AK/HS/Epcot.