We are basically doing what Edith describes, but we have set up an iTunes account for each team as they can choose to have different apps on their iPads. We also have a separate iTunes account for the Music cart and one for the cart that can be signed out.
We have set up each cart using Configurator, which we're regretting. Configurator was cludgy and difficult to use (try applying updates to apps), and lately has become completely broken. We have one cart where we can't apply license codes we've purchased and Apple tech support is pretty useless. At this point we're wishing we had stuck with the simple model of sharing an iTunes account and using the iTunes Wi-Fi Sync to distribute new apps to the iPads.
For the Wi-Fi Sync to work the iPads must be on and be signed into the iTunes account. This seems to not always be the case so the iPads need to be checked before you assume they all got the new app. It is easy enough to sign into the iTunes account and go to Purchased Apps and manually download the app. Of course we would use the volume purchase to remain legal, but only actually use one license.