Generally when you would want to subpoena an out-of-state person or documents you would want what is called a "Open Commission." This is where the Court where the matter is pending permits you to subpoena outside the state. Even if this is granted however, you still need to serve the party in accordance with the rules for service.
I can't tell whether you want documents or the person to appear. You can always send a subpoena. If the subject agrees to appear or produces what you want, you're done. No state court has jurisdiction beyond its borders so a supoena from one state cannot compel production or appearance from any entity outside its jurisdiction. If the subject does not voluntarily comply with the subpoena, you need to file an action in the local court which does have jurisdiction. This often means hiring a law firm in that jurisdiction to coordinate efforts. The local sheriff would then serve your subpoena. This is an expensive proposition. Don't forget you are subject to pay appearance fees and travel expense, in advance. The amount will probably be dictated by the rules in the juridiction where the subject is served. If you are going to this much effort and expense, you sould hire a lawyer.
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Normally you cannot simply serve a subpoena on a out-of-state witness. For example, if a California litigant wants to serve a deposition subpoena on a out-of-state witness, that litigant must file a motion in the pending California court for issuance of a "commission" to a deposition officer in the target state. Such appointment effectively authorizes that person to administer oaths and to take testimony for use in the California action. (A subpoena for records only will work in a similar manner.) After the commission has been issued, the California litigant will normally need to retain the services of a court reporter, attorney or judicial officer in the target state to serve the subpoena and obtain the desired information.
In regard to the affiliate that is doing business in Florida, I do not know whether that affiliate is authorized to accept service on behalf of the entity in Puerto Rico. I would guess not.
Disclaimer: This response is for information purpose only and does not constitute a legal advice. This response does not create an attorney-client relationship.