If it were me, I'd be looking at suction cups of some sort. Maybe some of these: 2 Acupuncture Massage Suction Cup Heat back Pain Relief | eBay but it might be difficult to remove the work piece once it was under vacuum.

The other option might be to use dent pullers like these: Mini Dent Puller Bodywork Panel Remover Removal Tool Car Van Suction Cup Glass | eBay but attach them to the machine and activate them with an air cylinder or some other mechanical actuator.

On the syringe theme, you could use air cylinders to create a vacuum. You could either use one air cylinder and actuate it manually or use 2+ with one as the actuator (i.e. you feed one with compressed air and it then pulls the other(s) to create a vacuum). Air cylinders are not cheap to buy new but can be manufactured at home if you have a decent lathe. Hydraulic cylinders are another option (minus the oil!). Or the scrap yard...