The answer is a resounding YES, regardless of which camera you own or use!
While you may use literally ANY camera, I primarily use a Canon DSLR (1ds2 and 5d2) with an assortment of prime lenses. I choose to use these because it gives me a lot of flexibility when shooting in a wide variety of lighting conditions. For example, I might start photographing my family in my home early in the morning where light is low, but the scene may shift quickly as they move outside. By using a prime lens with a wide aperture setting, I do not have to change lenses when my subjects move into different light.
With that said, ANY camera will work, but there just might be some limitations. My advice would be to understand and work within the confines of those limitations. For example, if you own a Canon DSLR and the standard 18-55mm kit lens, shooting inside without flash will be difficult. Therefore you might want to opt for more outside documentation. Or if you have an old manual focus film camera, you might want to photograph your family when the are more subdued and not participating in an action packed event.
While I do prefer my Canon DSLRs, I occasionally shoot other cameras. I owned two different Leica M6 film cameras with 35mm lenses and REALLY enjoyed them. One of the limitations of the Leica M is that the lenses are manual focus. Additionally, since it is a film camera, you cannot adjust your ISO on the fly. Posted below is one of my favorite sets from last summer. The photos were taken while we were on vacation. We decided to take a late evening beach walk. I only had about a half of the roll of film left in the Leica M6, so these are all the shots from the session. The film used was Fuji Pro 400
So, to recap, any camera will do! Now go photograph your family!