Based on my understanding, activating a Tentacle Rod requires the "Use an Object" action. As a 3rd level warlock with Pact of the Chain and an imp familiar, can I order my imp to carry and activate a Tentacle Rod to attack enemies?
The Tentacle Rod does not actually cause the wielder to make attacks. The tentacles attack, on the wielder's command:
While holding the rod, you can use an action to direct each tentacle to attack a creature you can see within 15 feet of you.
Note that it doesn't matter whether this is an Attack action, a Use an Object action, an item-specific action, or any other kind. Familiars can't attack using any action, or by any other means (such as opportunity attacks)*. For example, they couldn't cast green-flame blade, because it requires making a melee attack. But the Tentacle Rod is special: you're not attacking with it, just holding it and pointing at targets.
*Pact of the Chain familiars do get a narrow exception to that, but it's not relevant here.