The answer to your question would be that the pronoun that correctly completes the sentence using the objective case is the following one: Me. That is, the sentence would be as follows: Please leave the key with either Carol or me.
In this case, a pronoun in the objective case is needed. The pronoun in question functions as the object of the preposition "with". If the proper noun "Carol" was replaced, it would be changed by "her". This also indicates that a pronoun in the objective case is required after "or".