Pic Puller and ExpressionEngine Multiple Site Manager.
If you have ExpressionEngine Multiple Site Manager, you can use Pic Puller to manage multiple Instagram application from within a ExpressionEngine installation. Below are a series of screenshots showing how multiple apps are managed by Pic Puller.
Pic Puller's Review All Apps screen.
Pic Puller showing authorized ExpressionEngine users of Site A's Instagram application.
Pic Puller showing authorized ExpressionEngine users of Site B's Instagram application. These are completely unique from Site A's application.
In the scenario where you log in from SiteA but need to authorize SiteB's users, you use a front-end authorization screen on SiteB. You may set this in the control panel to make it easier for SiteB users to find the front-end authorization screen.