Does this exist?

Does anyone know of server-side software that accomplishes the same as the apps that allow Apple's mobile users to view flash on their respective devices?

I just read an article that listed the fiscal 2012 stats for those devices: 125M iPhones, 58.31M iPads

That's quite a few users that are prohibited from enjoying flash apps.

As challenging as it can be to get traffic in the first place, to then suggest a visitor leave your site to go purchase/download/install software just so they can use the features on your site is asking quite a lot, don't you think?

Realistic solution(s)?

I dont know of any modules that will do this, but have you thought about converting the content from flash to html using a converter? I have done this in the past with some degree of success.... try typing flash to html converter into google....

Nathan : )

 Thanks, Nathan.

I didn't think any module could accomplish that, but I was hoping that whatever software that is used in those apps would be available for licensing.

From what I've read thus far, most of the "one-click-and-you're-done" converters leaves a lot to be desired.
Google has "swiffy" that I just downloaded to try out, as an extension for Flash Professional. Adobe also has what appears to be the most promising in its CreateJS toolkit extension for Flash.  Although they claim a good amount of accuracy, it seems to require the .fla file to work with.

If one has access to that file, then it would probably be fun to play with that toolkit. But I think the only way to affect a .swf file is to use a de-compiler to strip it down. Then you're in copyright infringement territory...