Is it possible for an Android App to Auto-Install and Run an Update?

I have an application installed on phones sitting in locked boxes in different states. It runs 24/7 and has little to no user interaction.

I know how to push updates to the marketplace...can I have a thread running in my application that sees an app update, installs the apk, and opens the new apk, replacing the currently running version with it?

Shouldn't be a problem with 2.2, you cannot do with this 2.1 or older versions. You don't even need to run a thread, you can let the market updater handle it for you. You just need to initially set the app to auto-update. And as kcoppock said, you can't change permissions.