Short answer: no.
>"making Giclee reproductions"
At what size? Even a smallish 11x14 print @ 300 dpi/ppi is roughly 13 megapixels. (The math is: (width X dpi) X (height X dpi) so you can calculate what you need for larger print sizes.)
>"Is there a digital camera that would give me the quality that I need to do all this?"
Just the quality part, sure: Canon 1Ds, Nikon D3X, Sony A900...
>"In addition this would be used for the usual trip, family, etc. photos so would like a smallish camera."
You're not going to get the above level of quality in anything that I would consider smallish. (Or cheapish, but you don't say that budget is a factor.)
Given that, I would recommend you get two cameras: one for art and the other for family stuff.
I am wanting to buy a digital camera so I can take photos of my art. These images will be used for submitting to shows and I would also like to see about making Giclee reproductions of some of my pieces. In addition this would be used for the usual trip, family, etc. photos so would like a smallish camera.
Is there a digital camera that would give me the quality that I need to do all this? I am not a camera person and would like a simple one to use.
The photograher that I presently use to photograph my art is less available and I tend to have my paintings sold before getting them photographed by him.