I think the normal term is "champing".
Generally, far as I can tell, champing is where the mouth works and chews at either nothing, or at a specified something (chewing tobacco, gum, cigar, bridle-bit...) that will not be damaged or destroyed.
Unfortunately, while I can find quotes, I can't get a dictionary reference which doesn't include teeth, though the canonical "champing at the bit" clearly doesn't involve teeth, since a horse's bit goes into the mouth behind the horse's teeth.
he smacked his lips and champed his gums while muttering
-- The Scarlet Plague, by Jack London, 1915.
she was an old hag in a great tremendous poke-bonnet,
ambling along backroads at twilight and champing her gums
and spitting at people she passed.
-- God Bless the Devil, by James R Aswell, 1940.
A tropical field rat, gnawing a hole in a cardboard box, recently scampered off with Boardman's lower set of false teeth. "I'm in an awful fixth," the Seabee explained as he chomped-champed his gums. "What good will my uppersth be without th' lowersth?
-- Marine Corps Chevron, by Sgt. A. D. Hawkins, 25 Nov 1944
I gave only one look at the aged grandmother: wrinkled, bony, hunched, almost bald, toothlessly champing her granulated lips, her eyes red and gummy"
-- The Journeyer, by Gary Jennings, 1984.
Mother is there and eating well -- I know that, having seen her at it -- greedily sucking in her thin ragged lips, champing toothlessly away, her eyes dull and lustreless
-- The Time: Night, by Li͡udmila Petrushevskai͡a, Sally Laird, 1992.
an old mottled Chinese nonya champed her gums at the open door
-- The Long Day Wanes: A Malayan Trilogy, by Anthony Burgess, 1992.
The crone champed her gums together and nodded brusquely
-- Forged by Fire, by Janine Cross, 2008.
a toothless middle-aged lady whose wide mouth seemed to be the only mobile part of her enormous face. She had a habit of champing their lips while she sat silent
-- Dance Of The Apprentices, by Edward Gaitens, 2010.
The old woman suddenly lurched, champing her gums irritably.
-- Reading the Sauce Bottle, by Martin Stuart, 2013.