If you haven't done already, apply for the Adobe Flash Player Distribution Rights for your company https://distribute.adobe.com/mmform/index.cfm?name=distribution_form&pv=fp , then you get access to a special page. On this page you'll find all required Downloads for distribution inside a company.
There is also the "Background Updates Resources" archive you can extract this and put it on a web server.
Then the only thing that's left is to change the mms.cfg on the clients to point to this web server. You could use a machine login script, GP preferences or other deployment tools at your hand to get this file out to your clients.
It's all outlined in the guides on the distribution page.
https://www.adobe.com/content/dam/acom/en/devnet/flashplayer/articles/flash_player_admin_guide/pdf/flash_player_31_0_adm…
Page 28 "Backgroud Updates from an internal server"
Needless to say, that you have to download the Archive and extract it to your server every time a new Flash Player is published.
Best Regards
Mario