When can I do the supply missions?

Chronologically, the Supply Run missions are supposed to take place throughout Season One. You should be able to run them at any time. As this question is a bit dated, it's possible the Supply Runs were still in development or required a purchase when you asked the question, but they should now be unlocked and available for no additional charge. You should not have to complete the whole story before receiving access to the Supply Runs, but if they somehow still show up locked, you may have to update your app, run the first mission, or upgrade your base before you get access.