I did robots for elementary age kids for 3 summers now. Now, I only have them for one week of 2 hours a day, so an ongoing thing run as an after school program might work better, but based on my experience, I think it would be rather painful.
Right up front you need to understand that there's a big difference between working with your kid or your kid and their 2-3 friends and working with a room full of kids that don't really know you. Kids no longer have built-in respect for adults. They'll run right over you and treat you like a substitute teacher. It takes a lot of work to get and maintain control of a group of elementary age kids, especially around exciting stuff like robots. Parents being there is a mixed bag of stuff. Some are great and a few will make you nuts.
I've tried Lego Mindstorms and even just taking the top 2-3 kids and setting them up with a (original not NXT) Mindstorms kit didn't work. They didn't have the patience to follow build instructions for more than 15 minutes. They cobbled together some "stick a wheel on a motor" platforms and crashed them into each other. That's about as far as I could get them in 2 days so I've never gone back to Mindstorms in that group. I would never try Mindstorms with the large group of all the kids in our camp, unless we planned on the kits being "expendable". Too many pieces go missing.
I've tried Vex and that's worked somewhat well. I have seveal of the original Vex kits and we've only used them in radio-control mode. We've done mostly robo "sports" like soccer, obstacle courses, and "just smash into each other". Each kit doesn't come with enough to build much of an arm, but you can make a single servo flipper / weapon with something like a cardboard scoop. Or put a kill button on top and let them smack each other's kill button. I haven't had elementary kids building anything with Vex parts, but they did do cardboard work like scoops and "armor". The Vex kits held up well. One even got squashed hard by a table and I was able to hammer all the bent metal back mostly to the original shape.
There is a newer, significantly cheaper Vex kit that's mostly plastic and comes with a gripper arm and a wireless camera. If I were to go shopping now, that's what I'd try.
I've also tried teaching anything resembling programming, algorithms, etc. and have made no progress. I tried some activities from "Computer Science Unplugged", an ebook sold online. It's got hands-on activities for kids like having them bubble sort themselves. It looks neat, but in practice I had a real control problem when I tried it. It might work better in a more controlled, longer-term setting, like a classroom.
We also tried some scratch-built bots using cheap motors and eventually salvaged toys, but those never worked that well. If you don't have a geared-down motor there isn't much you can do. You can set up motors to just spin the shafts as the wheels so that slows them down, but you still basically have a dumb, fast bug that just runs. Adding logic and sensors to it is possible, but pretty far beyond anything I've been able to teach to kids that age.
We did watch the "Great Robot Race" DARPA challenge video from Nova. Two years in a row it kept the kids fairly captivated if I split it into 15 minute segments and we discussed it in between.
For year two, one of the activities we did was "Robot Arena 2", a PC game version BattleBots. You can do construction, including picking different weapons, wheels, motors, etc. It was a rough start, but once we had a few kids into it, it was a real winner. It runs on pretty old PCs and is only $20 if you can find it. Unfortunately we could never get network play working reliably.
Now, through all of this, I've had my own elementary-age daughter working on both Mindstorms and Vex. She's been perfectly capable of building stuff with Vex, following the Mindstorms builds, and understanding the algorithms for the built-in Mindstorms programs. I also know elementary age kids do First Lego League. So I know elementary-age kids can do it, but I haven't been able to make it work in the setting I've had.
You also need to manage the kids' (and especially their parents') expectations. The first year I did this, parents thought their kid was going to take home a robot. They paid $20 for the week and 100% of that went to tshirts, pizza, etc. and definitely not robots or batteries. The old Mindstorms cost $150 used. The Vex kits at the time cost $150 on a half-off sale. I had to explain the math to several sets of parents. I shopped hard for a "take home" robot kit for the next year and the best I could do for something reasonable that could be built by the kids with some help and would still do something interesting was $20-30.
But don't let me scare you off. Even though I think we've run a pretty crappy robot camp, the kids mostly loved it. I personally spent over $1000 on Mindstorms and Vex just to do the camp and another volunteer spent at least a few hundred dollars too. Despite all our frustrations, we haven't really regretted doing it. I'd love to do a robotics class at our local homeschool co-op or do some sort of after-school program for kids that are really into it, but can't make the schedule work.