dnd 5e – Can an ally or familiar with opposable thumbs (Imp, Sprite) doff and/or don the player’s shield for them?

The short answer is that it is up to the DM. I couldn’t find any supporting rules specifically about getying help into or out of armor. However….
Climb onto bigger creatures

As an alternative, a suitably large opponent can be treated as terrain for the purpose of jumping onto its back or clinging to a limb.

This rule allows small creatures to climb onto you and at least attempt to buckle the shield in place.

Doff and Don armor
Says:

Don. This is the time it takes to put on the item. You benefit from it’s AC only if you take the full time to don it.

Doff. This is the time it takes to take off the item. If you have help removing armor, reduce this time by half.

So at the very least they can help.