Specific prebuilt Flash players with decent reputations include
https://flowplayer.org/ and https://www.longtailvideo.com/players/
It might be worth starting with the longtail player.
There’s also https://downloadsquad.switched.com/2010/04/10/sublimevideo-html5-video-player-adds-flash-support/ which might be worth exploring : or search for a Flash player with HTML5 / iOS support.
Otherwise, you are entering a hot zone. Flash has been the dominant force in web video for some time, using different codecs inside a Flash video wrapper. This provides great flexibility and programmability, and works on nearly all desktop platforms.
But in recent times Apple has sold a lot of iPhones and iPads (with good web browsing capability), and does not support Flash video on these devices.
So as of today, unless you can find a way of combining Flash and iOS support, you are probably looking at having at least two solutions : something for Apple using H.264 encoding and no specific video player, relying on HTML5 video playback on Apple devices, and something using a Flash player for the users without HTML 5 support – likely to be the majority, for the next period, as Microsoft has no plans to support HTML 5 except in the almost-released IE 9 on Windows 7 and Vista. Windows XP is likely to be around for a good while yet (there’s still a sizeable proportion of people on the web relying on IE6).
HTML 5 is nowhere near an agreed or finalised specification, and so browsers billed as HTML 5 today can and do support different subsets of the provisional HTML 5 spec in different ways. There’s no agreement on potential video codecs support in HTML 5, with Apple and Microsoft favouring H.264 (on which they receive royalty payments) and Firefox, Chrome, Opera and others favouring WebM. This is unfortunate for developers looking for a neat, single solution.
Video for Everybody is another approach to resolving this : https://camendesign.com/code/video_for_everybody
This might be helpful too https://www.html5video.org/demos/