Juan,
The management company works for the Board. Therefore, the Board may request anything relating to the Association. However, I would expect that the management companies computers belong to the MC, therefore, they probably don't have to provide any personal emails. If the e-mails are considerd personal vs Association you probably won't get them without a court order.
Per your
previous thread in this forum, I had already recommended that the Board review the terms of the contract with your MC and possibly terminate the contract. At the very least, the Board should open the contract up for bids when it's time to renew and, if it were me - I would strongly consider not awarding the contact to the existing MC no matter what the savings might be.
Tim