No, swamp coolers don’t require venting as air conditioners do. But, you will need to take care of the excess air pushed out of the system. If you don't, your room might get stuffy. Usually, in such cases, opening a window or two will do.
How much ventilation you need also depends on the size of your room compared to the size of your swamp cooler. Normally, if your swamp cooler gives out about 1000 CFM, you’ll need about two square foot opening to facilitate the ventilation needed.