I think you have two strategies.
First, if you only need some of your things in Winter Park and you can arrange your luggage accordingly, check what you don't need in Winter Park to Reno when you check in at Chicago; it will be stored until you actually arrive in Reno on the second leg of your trip. Carry what you will need in Winter Park with you on the train. As others have said, larger suitcases can be stored in the public luggage rack on the bottom level of the Superliner car. Since you mention staying in a Roomette, there's very little room for two people plus any amount of luggage beyond a backpack each, you will probably be storing your luggage in the public rack anyway. Even when I'm by myself in a Roomette, I only take what I need for the night with me to the Roomette and leave my other luggage in the public rack downstairs because maneuvering a suitcase up/down the stairs isn't all that elegant or easy.
Second option is to carry all of your luggage on the train with you but do know that there is only so much public rack space in each Superliner car and it is first come, first served, so to speak. If you go this route, I'd arrange for red cap assistance to get you from the Chicago Metropolitan Lounge to the train with all of your luggage, which may get you to the car ahead of others who aren't requesting red cap assistance. If the red cap attendant is particularly helpful in the endeavor, remember to offer a nice tip.