<p>It’s “their” because “chewing” is a gerund–a form of a verb being used as a noun. Therefore, “chewing” takes the adjectival form “their” as its modifier.</p>
<p>Cr7_ManUtd would have a valid point about the acceptability of “them,” if the sentence ended with the word “wood.” But it doesn’t work with the "is sometimes . . . "</p>
<p>Edit: Wow, I must have hit some weird control key. Sorry about the repeated sentence and the incomprehensible part–can’t find the hidden “glyph of incomprehensibility” that caused it.</p>
<p>Well, yes, 112358, you could say
. . . damage caused by them, chewing through wood, is sometimes visible . . .
but then you’d need 2 commas to set off the participial phrase.</p>
<p>CB does use this type of question, with the possessive modifying a gerund, quite a lot.</p>
<p>Having realized that the damage is caused not by termites, but only by their chewing, I have decided to start a sideline business: QuantMech’s A-OK Termite Muzzles! :)</p>