Most digital cameras have a simple lens and CCD behind them. There are no mirrors involved in these cameras. A digital camera (and similar video cameras) would likely (but not necessarily) capture the image of a vampire.
Going off subject to film cameras we get a rather different issue that becomes interesting to explore (that does have some repercussions back upon the digital camera).
First the type of camera makes a difference. A simple point and shoot camera like the disposable Kodak cameras would have no problems at all as they do not involve any mirrors. In most of these cases, the optics is a simple hole in the box that approximates the viewing area (leading to parallax error, but that has nothing to do with vampires) and anywhere from a plastic aspherical lens element to maybe two or three glass elements. No mirrors.
However, the TLR (Twin Lens Reflex) and SLR (Single Lens Reflex) cameras involve a mirror - that is what the "R" stands for - reflex. This would reveal itself as you just can't see the vampire through the camera making it very difficult to focus on the vampire. Realize, this makes the use of telescopes and telephoto reflex lenses right out of the question too (the moon could be populated by vampires and we could never see them through a modern reflector telescope - have to go back and check with the refractors). A view camera, being a very large point and shoot would allow for this. Sharq notes that the older 6x6 Polaroid type cameras do use a mirror between the lens and the film for enlargement of the image onto the film. Meanwhile, the newer Polaroid cameras use a drastically smaller film. Once again, there is a 'however' that brings us to the second issue.
Silver halide is used in all types of film (and glass plates). When exposed to light, the silver halide crystals become reduced (as opposed to oxidized). From this it can safely be assumed that a vampire's image cannot be captured on film and that all movies that include vampires are done by actors who have some soul (they may have sold it, but there is a soul somewhere).
Interestingly, at the dawn of photography silver halide was not used to capture the light. Silver halide was first used in 1839 while the first pictures were taken in 1826. These early photographs used asphalt as the "film" - though it took about 5 days to properly expose the "film". In theory, this early photography could capture a vampire's image if the vampire was cooperative enough to lie in the sun for 5 days. Clearly, given the subject, this is impractical. There may be some films where a different compound is used for sensitivity in other areas of the spectrum such as infrared and ultraviolet. However, I am uncertain if this is the case (and suspect it is still silver based). Not to mention the fact that vampires are rather cold being undead - the best bet is ultraviolet. The difficulty with ultraviolet is that it is not transmitted by glass and would require a special (and expensive) lens.
Returning back to the issue of vampires and digital photography, there is the question as to if this is silver, or if any reflective metallic surface will work. If this is not the case, then it seems rather unlikely that a digital camera, filled with shining thing-er-magis will be able to record a vampire's presence. Even if it was, gold and silver are often used in electronics for their excellent conductivity (especialy gold which does not oxidize). Silver may very well be used in the CCDs too.
Digital SLR user and photographer.