Most of the time, yes, you can go to 4 High in motion. Even my 1995 Explorer with an electronically selectable t-case could be switched into 4 Auto or 4 High while in motion, but it had to be below a certain speed. 4 Low required stopping, and probably does on most/all systems.
I'm not sure if my 1994 Ranger with manual t-case can go into 4 High while in motion, and I haven't tried, but it seems like it probably could. The shifter isn't gated from 2 High to 4 High, you just push it forward, and there is a gated neutral position before you can get to 4 Low.