Track down an air charter service with a Stearman in the fleet. Call and discuss charter fees and book a flight. Tell the agent and pilot of your desire to take the controls for a few moments during the flight. One such organization is Tailwheels, Inc. based out of Winter Haven, Florida. Use the AOPA website to locate Stearman centers.
Enroll in lessons in a Stearman to get the necessary "type certificate endorsement" for your pilot's license and logbook if you are an existing pilot. As a general rule, it takes about four to six hours to get the needed type endorsement for the Stearman. This let's you rent or operate a Stearman throughout the nation as you wish (rent from the various locations.)
Go to a fly-in aviation weekend. One such fly-in (and one of the largest in the nation) is the EEA OshKosh AirVenture Fly-in in Wisconsin. Find Stearman owners and talk them into taking you up for a flight, not a hard thing to do at events such as these (aviators love to show off their airplanes.)