You will (ideally!) need to be able to have the original ActionScript sources for your Flash project.
Haxelib
ActionScript 3.0 can be converted using the free “as3hx” tool, or by hand, into the Haxe language. Then many of the original Flash features will continue to work in OpenFL, but the code will compile directly to C++, back to a Flash SWF, or to JavaScript. This is how most developers ship their live products for HTML5, native desktop, mobile and console platforms.
NPM
ActionScript 3.0 can be used with the Apache Royale project, to transpile to JavaScript. This also pairs with the OpenFL project to create an HTML5-only project, though I don’t know anyone who has shipped with this yet, as Apache Royale on NPM is new.
ActionScript 3.0 code could also be converted by hand to JavaScript or to TypeScript, or converted (like above) to Haxe, then linked with the OpenFL library for HTML5 deployment.
SWFs
A final Flash SWF may contain fonts, graphics, sounds and compiled ActionScript code. We currently have support for most SWF graphics in OpenFL, with ways to link it to your code. The Shumway project is the most mature solution I know to play a SWF file as-is (with no modifications), but there are some serious problems with that approach. Performance is an issue, but shipping real products often requires changes to your code.
Shumway
The Shumway project has been abandoned, but perhaps there is a fork with more promise.
Here’s a blog post from someone who tried to use Shumway recently, there may be other stories of success or failure: https://dev.to/elmuerte/shumway-shiv-scant-success-o1a
Next Steps
Going back to see if you have source code for your Flash SWF project will go a long way to helping know what your options are.
A SWF decompiler might be able to help, but converting will probably require some work, though certainly less than rewriting in JavaScript and having no Flash API library