But I'm still not sure if I should begin my email to the US Consular Agency here in Spain as "Dear Consular Agency," because I don't know who within the agency will handle my email.

Hello Lynn, The person writing under the name of "cool" is apparently not a native English speaker, because he has reversed the order of the noun and the adjective in "personal hiring" from the way we say it in English and he has used the word "personal" instead of "personnel" as one would do in Spanish and a few other Romance languages. I think that what cool meant to ask is if he may use the salutation "Hiring Personnel." It is probably worth noting too that in Spanish, only the first word in the salutation gets capitalized; therefore, cool should be aware that his question should have been if he may use the salutation "Hiring Personnel," with both words capitalized. Since the question was addressed to you, not me, I'll let you take it from there.