As I see it you have two basic options.
1. Refactor your extension to remove the jQuery dependency.
2. Continue using jQuery and shim the DOM APIs that it requires
Personally I'd lean towards removing the jQuery dependency, but depending on how heavily it's being used that may be a lot of work. If you go the shim route, try looking for resources that describe how to use jQuery inside non-document JavaScript contexts, like this blog post about
using jQuery inside a WebWorker. Alternatively, you could try using a DOM library to shim the missing document object. I'd probably start by experimenting with
linkedom, but note that that library doesn't come in a nicely packaged, single-file format.