I'm looking for a web-based application that I can install on a
website that will let people design the layout of a room.  This will
ideally be something interactive, like Flash or ActiveX, where people
can drag and drop common room elements (e.g. chair, TV) onto a grid
that is shaped like their room, and can help them to design their
room.  I'm happy to pay for this application, if needed.

This type of program can be very complicated, and therefore very
expensive. As far as I know this is a very customized program since
you want to integrate it into your website. There are many software
programs you can buy to do this offline at a consumer level or in a
professional CAD program. Do you have a library of objects users
(graphics of furniture for instance) will use to put in the room, or
do you need those to be created too? What is your goal for users
arranging these rooms? Will it be part of your sales process? Will
they need to print it out?

You might try here to see if there is a programmer available for hire
- maybe a freelancer.  I've visited here before and have seen
favorable reviews though I haven't had a need for it yet....

http://www.2rentacoder.com/RentACoder/default.asp

-Hope this helps

Re: 2rentacoder.com... terrific idea!  I'll check it out!