Yes, many of the point and shoots offer a tethered mode. I'm most familiar with this feature in the Canon Powershot line of P/S cameras. The camera comes with software to do this.

There are a few more factors here to consider besides just brand names. If the camera can be run from an AC adapter that is a huge help in doing this type of photography for more that a few shots. Chances are you'll have the camera on and focusing a lot and be using flash. This type of work eats batteries or rechargeable cells. Make sure that the camera you purchase has a tripod mount. Not all do, as many are just too thin or compact to allow for one. A remote is handy to trigger the camera, but not an absolute necessity. Be sure to have at least two memory cards if you plan on swapping them out to print. Probably be good to have three just in case of backups while printing and depending on the size of the card and how many photos you plan to take. They don't have to be the biggest size in this case, the sweet-spot for flash card memory prices is falling in the 256-512 M size it seems to me right now. Don't forget the tripod cost as part of your calculation unless you have one. For this kind of thing you can probably use a very inexpensive video type tripod and be fine. Just make sure it is stable and/or weighted down if not on a concrete floor.

I'm not sure of your budget range?

Around $200 plus ink and paper -
You can look at photo combination packages from Lexmark or the like that have a camera and printer in the box. Just make sure you get extra batteries, paper and ink. Not all of those are the easiest to use as they may not allow the camera to be directly hooked to the printer.

$300-500 depending on printer plus ink and paper -
As an example of a possibility here is the Canon Powershot S2-IS which has Image Stabilization as well. This really helps in making a good photograph and fixes a lot of "shaky" hand issues. This is a PictBridge enabled camera which allows it to plug directly into an enabled printer and then you use the camera display to set up the printing.Canon has some dedicated photo printers, but calculate your costs as these can be pricey per print.
There are good units from really all the major names, Nikon, Olmpus, Kodak, Fuji and so on besides Canon. Printers from Epson or H-P would complement this range.

Probably going to the local Electronics superstore and seeing what you going to spend (for everything) and then check the reviews here at http://www.photographyreview.com/reviewscrx.aspx
for any additional comments or hidden information. It also is a good idea to see if the camera looks "friendly" and won't be too trick to teach several people to use properly.

The real cost in all this is the paper and ink. That the big thing to consider when buying the printer is what is the per photo cost. DyeSub printers are fairly easy to calculate as they usually cost x amount per image for paper and ribbon. Ink-Jets can vary tremendously on paper costs and how much ink the paper soaks up to look good.

Hope this helps, if you have more questions please ask. Somebody here will have an answer or input I know!