R.I. Gen. Laws § 10-3-6: Judicial appointment of arbitrators.

If, in the agreement, provision is made for a method of naming or appointing an arbitrator or arbitrators or an umpire, the method shall be followed; but if no method is provided in the agreement, or if a method is provided and any party thereto shall fail to avail himself or herself of such method, or if for any other reason there shall be a lapse in the naming of an arbitrator or arbitrators or an umpire, or in filling a vacancy, then, upon the application of either party to the controversy, the court, as described in § 10-3-4, shall designate and appoint an arbitrator or arbitrators or umpire, as the case may require, who shall act under the agreement with the same force and effect as if he, she, or they had been specifically named in the agreement; and, unless otherwise provided in the agreement, the arbitration shall be by a single arbitrator.