Photographing a bright object is easy. However, photographing a faint object is much more difficult because of Earth's rotation. The sky rotates 360° in 24 hrs, which boils down to 1/4° (15 arcmin) per minute. This probably doesn't mean much to you, but it's the same as saying that if you set up your camera to point directly at a star, over the course of 1 minute, that star will move 1/2 the width of the full moon. If you are trying to image something like M31 (Andromeda Galaxy), then you'll probably have a lens on with a ~6° field of view. If you're using - for example - a Canon 350D that's about 3500 px across, then the galaxy would move by approximately 150 pixels over the course of a 1-minute exposure. Andromeda is very bright, but its light is spread across an area over 40 times the size of the full moon which effectively decreases its brightness by a factor of 10,000; so a short enough shot without a tracking mount will result in black.
So, without a mount that tracks with Earth's rotation, you are limited to very bright objects (e.g., Moon, Galilean Satellite positions around Jupiter, very bright comets (rare)) or to objects where you want it to be blurred across the frame (like star trails). But if you have a mount that tracks, then you are only limited by the size of your lens (aperture and length).
Thus, to directly answer the thread's title: Yes, you can do astrophotography without a telescope. A telescope just acts to collect much more light (light collection goes as area of the lens/mirror) than a camera lens, and they usually have built-in clock drives to compensate for Earth's motion.