Are there any specific type of FDM 3D printers that I should look for?
Not all printers are suitable to print flexible filament. E.g. 1.75 mm filament printers with a Bowden extruder/hotend combination will not work (you may have more luck using 2.85 mm filament, which is stiffer because of the increased diameter). For 1.75 mm filament you require a direct drive extruder, e.g. with the stepper mounted onto the hotend, even then some additional guide parts need to be printed to make it work. This also depends on the amount of flexibility of the filament, some are more flexible than others.
E.g. Ultimaker 3D printers use 2.85 mm filament with a Bowden setup. They also sell a flexible filament that can be printed with these printers. Even for direct drive extruder printers like the Anet A8 (a cheap Chinese Prusa i3 clone) inserts exist (e.g. this or this one) to even better guide the filament to prevent it to buckle.