We have done it a couple of times. No kids. Two adults. Used Disney transportation exclusively.
We do AK first, since it's the most off on it's own, taking the bus from resort. Then bus to HS, then boat to Epcot, finally monorail to MK. We haven't done this since the Skyliner opened, but I think we still would do things this way. MK is usually open the latest, so we like to end there. When we have done it, we have done our pre-booked fastpasses at HS and then as soon as we tap into our last FP there, start looking to see what we can get at the next park.
Trying to accomplish SDD, TSMM, 7DMT, and PP on a 4 parks one day day may prove pretty challenging though. Without FPs, those all have some pretty long wait times, and they are also difficult to get as a same-day FP (especially SDD and 7D). I'm sure it's not impossible, but it might prove to be a challenge.
Here's what we have done on our previous 4 parks 1 day days:
2017
AK - Dinosaur, EE, meet Donald Duck, photo with Tree of Life
HS - ToT, TSM, meet Donald Duck, photo with ToT
Epcot - FEA, GFT, Soarin', SSE, meet Donald Duck, photo with the ball
MK - Tiki Room, PotC, WDWRR, Philharmagic, Carousel, HM(x2), Magic Carpets, Buzz Space Ranger Spin, meet Donald Duck, photo with the castle
2018
AK - Navi, Dinosaur, meet Minnie & Mickey, photo with Tree of Life
HS - SDD, ToT(x3), meet Minnie & Mickey, photo with ToT
Epcot - France movie, Canada movie, Soarin', Living with the Land, SSE, meet Minnie & Mickey, photo with the ball
MK - HM, Small World, Splash, PotC, Buzz Space Ranger Spin, meet Mickey (we messed up the timing and missed the opportunity to get Minnie here), photo with the castle