I am a Warlock with an Imp familiar. Can I send my familiar to cast Mending on a mummy with the intent to have all or some of the torn wrappings be mended and therefore restrain, blind, or just distract it?
Short answer, no
One, familiars generally don't cast spells. The owner casts the spell and it can be delivered by a familiar. That will be in a different point. Imps, can polymorph and turn invisible, but they don't have the Mending cantrip (unless homebrewed for some reason). Therefore I have to assume you mean that the magic user casts Mending, and it will delivered via familiar.
Two, Mending takes 1 minute to cast. If it was an action, then you might have a chance. But this will take 10 rounds to complete. That's 10 rounds while in combat with the Mummy (you can't cast it as a Readied spell). Which means depending on how your DM rules it, 10 rounds where you can take no other action and at the end, spend the 11th round delivering the spell, -or- 10 rounds with your Imp in constant contact with the Mummy with the spell going off at the end.
And while combat with a Mummy can take a while, there are much better ways to use your time.
Three, nowhere in the rules does it suggest that Mending would somehow bind up a Mummy. Per the rules of Mending, it "repairs a single break or tear" so only one loop of cloth would be stitched back together.
Unless you could somehow specify exactly which tear would be the one that trips the Mummy up, it would be kind of pointless.