Flash is actually closed source and Adobe don't want to release it for ARM-powered devices (read: anything that is handheld).
They have been known to release versions of it for particular phones and so on, but only when paid several million dollars.
Basically, flash on the DS is not very likely.
However, as a solution to the OP's problem, you can listen to online radio that uses .m3u playlists in DS-Organize! Just find a nice online radio station and see if you can find their m3u playlist file.
I listen to Idobi!
Sorry, no DS web browser that supports flash. But, maybe you can create a file (or request one) that converts flash into a file that supports the DS carts...
Some attempted to make one, but I have no idea what were the results...