You need to review the rules for the specific type of arbitration you are involved in, to see how you can obtain documents from a third party. It normally involves the issuance and service of a subpoena. I do not see how you can issue a demand for production to anyone who is not a party to the arbitration.
Note that the discovery rules for arbitration are much, much different from those in the courts. you need to read the AAA rules carefully here.
Generally, after a case is commenced with the American Arbitration Association, you will have a telephonic conference with the arbitrator and all other parties/counsel in the case. You will discuss various discovery issues. You said that you were ordered into arbitration based upon a contractual agreement. That same agreement may contain a provision with respect to what type of discovery you are entitled to pursue. Generally, however, in an arbitration proceeding you would be able to subpoena documents from other parties and non-parties to the arbitration. You are correct that with the AAA you would obtain the subpoenas from the AAA. Unless you reach an agreement with the various attorneys for the non-parties, you could not simply serve the attorney for that non-party unless the attorney is the listed as the authorized agent for service. In addition, the individuals, or entities, who are defendants in the stayed state court action are non-parties in the AAA action so they would be treated precisely as non-parties. You can't simply issue them a Request for Production of Documents.
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As my esteemed colleagues above have stated, after the arbitrator(s) have been appointed, the pre-hearing conference will go over discovery issues, dates and other procedural matters. Arbitration in general has more limited discovery and your "alleged contractual" arbitration agreement may further limit discovery, but as you are dealing with a non-party to the agreement, the agreement can't, by itself, obligate a non-signatory, party or beneficiary beneficiary of the agreement. Thus, unless you were able to acquire the documents by request, you would need to acquire a subpoena from AAA arbitrator.
However, a common issue with non-party production in arbitration is not necessary the issuance of the subpoena (assuming your request is reasonably related to the matter, reasonably limited in scope, etc.), but enforcement of the subpoena assuming the non-party/third-party fails to comply with it. At that point, you generally must go back to the court to seek a motion to compel production, either under the FAA (federal) or the state arbitration act (for example, Cal. Code Civ. Pro. 1280-1294.2). The enforcement provisions are different under each and I would highly recommend you seek out competent counsel to protect your interests in the arbitration.
This answer was provided for general informational purposes only, and should not be construed or interpreted as providing legal advice or as creating an attorney-client relationship. I am licensed in the State of California. The law in your state may differ and your best answer will always come from a local attorney in your state that you meet and discuss the important facts related to your particular matter.