Do digital cameras have shutters

Short answer: Yes, if you define "shutter" widely enough.

Long answer: Rather than a physical shutter as one would have with a film camera, many digital cameras simply turn the sensor on and off as needed (called electronic shutter). This is usually used in point-and-shoot digital cameras. A digital SLR camera, on the other hand, will usually have a physical shutter which opens for the needed duration, just like a film camera. Some SLR cameras, like the Nikon D40 (and probably many others) use both a physical shutter and an electronic shutter; having the latter permits almost unlimited flash sync speed.