Flash Player End of Life

The specific date for Adobe Flash Player's End Of Life is December 20th 2020

Essentially, all major browser vendors will stop supporting Flash at the end 2020.  Please refer to Adobe's official announcement at Flash & The Future of Interactive Content | Adobe Blog​, which contains links to posts from the various browser vendors, who posted similar announcements the same day as Adobe's announcement.

In addition to the various browser vendor links in the above Adobe blog post, Chrome and Firefox has published Flash roadmaps detailing the changes they are actively making on the road to the end of 2020:

• Chrome: Flash Roadmap - The Chromium Projects
  
• Firefox: https://blog.mozilla.org/futurereleases/2017/07/25/firefox-roadmap-flash-end-life/  and Plugin Roadmap for Firefox - Plugins | MDN

Essentially I'm looking to know if we can continue to support an app written in Adobe Flex beyond the end of 2020 by using an older version of a browser (as a last resort) and if so which browser & version.

This really isn't recommended as it will require using a Flash Player version that isn't maintained or updated with security fixes.  It's safe to assume malicious actors will continue to exploit Flash knowing there won't be a security fix (after 2020) for whatever it is they find, or create to exploit Flash.  Assuming you'd want to take the risk, it's virtually impossible to guess what browser/version combination to use as browser vendors are continuously introducing end-user friction in the Flash space by being more strict with blocking Flash, forcing click-to-play for each domain (including sub-domains) and browsing sessions.