Disney is very bag friendly I know at least at EPCOT and MK. No need for lockers. I never been to AK so I can't say for them. HS I know RR you probably might need a bag for since it's a coaster that does barrel rolls, or leave it with a guest who might not want to ride. I do not know about TOT though since I never been on it.
As for Universal, as others said, if it requires you to stow it away period, they give it to you for free for the duration of your ride. Some rides you even need to stow everything on you. I've heard some people might use Child Swap as kind of a Bag Swap in the case of items that you cannot stow away (or perhaps if you do not want to use a locker you can try it as well?). However, if you go on water rides, they have lockers nearby, but unfortunately you do have to pay for them. So, if you do not want to do that, get your backpack a poncho? All joking aside, there are more bigger lockers as Mac said towards the front of the park.
For more details on which rides you need to stow away your bags on, you can find them online. I know the rides off the top of my head you need to store everything period (including what's on you) on the top of my head and that's The Hulk, Rip It Rocket, The Dragon Challenge.