You could try to find a way to convert it to Flash :-)
You could find a way to offer a VPN-type access to the site server, if it is Windows and can run the EXE file-type, then allow them to play it on the server ... but I wouldn't recommend it, if it can be done.
So option A above, or none at all is my best guess.
Yes it can be done without dowload If the webserver is php installed & allowing certain functions to be called. You can use a php script with the right functions to execute a exe file all within a html page
@Shnackhag, Hmm... I think your definition for running EXE is different from what it should be here. Don't think a browser would allow you to just take whatever content from running a exe from the server. Running exe from the server is one thing, but browser accept and stream it out as running is another. If you are talking about IE, may be able to but that's why IE is so prone of hacking. Other browser, I would expect it to ask you to download it first before you can run it. Still, it is obvious about security issue. Also PHP is an easy language for web but those who aren't careful could get the website into trouble -- can easily create a lot of security holes.