This question comes from a discussion with a DM friend of mine, whose players asked him if they could take a short rest while traveling in a moving cart. His initial reaction was to think that wouldn't work, but upon considering the rules for short rests:
A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
There doesn't seem to be anything explicitly prohibiting it there. One could argue that driving the cart could be considered a strenuous activity, but doing nothing more than sitting in the back seems to be a harder to justify as such.
Taking it a step further, we also considered the possibility of a long rest. Looking at the rules for long rests:
A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch. If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it.
We again considered that sitting in moving vehicle isn't explicitly called out, and intuitively doesn't seem to qualify as a "period of strenuous activity". Assuming one could rest on a sailing adventure (with the ship being a vehicle), and suddenly this seems less absurd.
And yet it's a concept that appears to go against the norm, with stopping and finding a place to specifically rest or set up camp, and I haven't seen the questions asked here or elsewhere. Is there something else written anywhere that prohibits this, or is our interpretation here wrong?