We would strongly discourage you from just running an unpatched Flash Player in an old browser. That's a pretty terrible plan.
That said, enterprises have managed to make legacy internal systems securely available to modern clients for as long as enterprise computing has been a thing, and given thoughtful design and implementation, you could engineer a way to do that safely. It would require a non-trivial amount of investment, expertise and effort.
There are also options to license (for a fee) maintained versions of Flash Player for use in your organization through our partner HARMAN moving forward. This might be a reasonable and less expensive path forward, and the folks at HARMAN would be happy to talk to you about your specific situation and make a recommendation.
Here's our official guidance to enterprise, which details a few of your options:
https://www.adobe.com/products/flashplayer/enterprise-end-of-life.html