Yes, a Muggle can do something with a wand.
According to the brand new "History of American Magic" writings on Pottermore, a Muggle (from a family described as being descended from wizards, but possessing no magic) was able to use a wand in a semi-controlled fashion:
Bartholomew had disseminated his leaflets widely, and a few newspapers
had taken him seriously enough to print pictures of Dorcus’s wand and
note that it ‘had a kick like a mule’ if waved.
We have a similar quote from Ilvermorny School of Witchcraft and Wizardry. Note that the wand rebels against use by a Muggle.
As Isolt watched, James finished marking the graves he had dug by hand, then picked up the two broken wands that had lain beside the
Boot parents. Frowning he examined the sparking core of dragon
heartstring that protruded from Mr Boot’s, then gave it a casual wave.
As invariably happens when a No-Maj waves a wand, it rebelled. James
was sent flying backwards across the clearing, hit a tree and was
knocked out cold.
We also see a movie example of a non-wizard (in this case, a monkey) using a wand in Fantastic Beasts and Where to Find Them.
JKR addressed this point in an interview in 2006;
I been asked what would happen if a Muggle picked up a magic wand in my world. And the answer would probably be something accidental... possibly quite violent. Because a wand, in my world, is merely a vehicle — a vessel for what lies inside the person.
and again in the footnotes for "The Tales of Beedle the Bard":
While the “rogue” ability to perform magic sometimes appears in those
of apparent non-magical descent (though several later studies have
suggested that there will have been a witch or wizard somewhere on the
family tree), Muggles cannot perform magic. The best — or worst — they
could hope for are random and uncontrollable effects generated by a
genuine magical wand, which, as an instrument through which magic is
supposed to be channeled, sometimes holds residual power, which it may
discharge at odd moments
This makes it clear that that while a true Muggle couldn't generally create a controllable magical force such as a spell or a hex through their own magical powers, they could potentially create a big bang or a explosion from the magic remaining within a magic wand.
There's also the (evidently quite small, but still real) possibility that they have sufficient magical heritage to make a wand do something.