According to Mike Mearls, yes.
Sage Advice: Can you short rest while in Druid Wild Shape and gain benefits of short rest?
The original tweet asking the question seems to have been deleted, but assuming the sageadvice.eu page is titled accurately, Mike Mearls says that he believes that this is indeed possible.
According to Jeremy Crawford, more generally:
Any creature, including undead and constructs, can use its HD during a short rest to regain hit points.
and
Any creature, including undead and constructs, can take a long rest to regain all its hit points.
Per PHB p. 186:
A short rest is a period of downtime, at least 1 hour long, where the character does nothing more strenuous than eating, drinking, reading or tending to wounds.
Nothing prevents a wild-shaped druid from fulfilling the requirements of a short rest. Remaining in Wild Shape is not described as requiring any strenuous effort.
Furthermore, per the description of the Wild Shape class feature:
When you transform, you assume the beast's hit points and Hit Dice.
Ostensibly, if you wanted, you (as the wild-shaped druid) could use the Hit Dice of your beast form to heal during your short rest. Mike Mearls agrees with this interpretation.
It isn't specified explicitly, but presumably your usage of these Hit Dice would not affect the Hit Dice of your regular form when you reverted out of Wild Shape. (The answers in this RPG.SE thread also seem to support this interpretation.)
And since Wild Shape is a class feature that recharges on a short rest, and since the feature description also states this:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can't use any of your special senses, such as darkvision, unless your new form also has that sense.
...then presumably you would also regain any expended uses of Wild Shape following the short rest. (The answers in this RPG.SE thread seem to support this as well.)
