Is there an app that can capture live video on one iOS device and broadcast it over wifi so all my other iOS devices can play it? Streaming it over the internet would be even better...
You can stream to other devices. This app does a lot, with video including uploading and sharing video on facebook, twitter etc but to stream video all the devices will need this app.
It also works across platform.
As far as personal experience goes I have only used it once to test it quickly, it workked accross a home WiFi network, other than curiosity I had no need for this.
You may be interested in this incredibly awesome little app called CollabraCam. I've never used it, but the idea is just brilliant. One iOS device serves as the "director," and receives video from up to 4 other iOS devices. The Director can send requests to the cameramen (pan up, shift left, zoom in), and I think can record video by switching between different live feeds.
CollabraCam is obviously designed for making movies with multiple camera operators, but it sounds like it might suit your simpler purpose of just broadcasting a live camera from one iOS device to another.
I should mention that I have absolutely no experience with this thing, I just think it's incredibly cool.
(By the way, the wording of your question ("capture live video") is a little confusing. I'm not sure if you want to capture video and stream it afterwards, or if you want to view a live feed with/without recording it. In any event, I think readers who arrive at your question because they are curious about streaming live video from one device to another will find CollabraCam interesting).
UPDATE: After answering this post I became curious and bought CollabraCam. I thought it would be fun to mount my iPhone on an RC car, and stream video back to my iPad. Turns out CollabraCam does work for live streaming, but not very fluidly. The video feed staggers a lot, and is definitely too slow to use for steering an RC car.
I didn't try anything to speed up the system -- I think I could create an ad hoc network with a laptop, and thereby get faster WiFi streaming than going over the home router...
But the short message is, CollabraCam does work for live streaming, but maybe there are better solutions out there.