Can the executor of a trust appoint a power of Attorney?

Executors normally concern themselves with estates, not with trusts. I think the party you're concerned with is a trustee.

A trustee of a trust should not be permitted to grant a power of attorney for his or her responsibilities under the trust, UNLESS state law specifically allows such an appointment, AND the trust specifically spells out the power to make such an appointment.

And, for what it's worth, an executor of an estate would be similarly prevented from appointing an attorney-in-fact.

In summary: No, fiduciary appointments are personal and cannot be delegated to third parties.