Currently,there is no Flash on the DSi. The DSi does not support Adobe Flash and there is no way to install it.
The idea of using an SD card has been floating around for a while now, but nothing verifies this and Adobe's download page has no Nintendo DSi listing. Programs cannot be run from SD Cards anyway, which is why an AceKard 2i is needed to play homebrews (DS software provided for free on the internet). There is no Adobe Flash "homebrew" (it would be illegal, as Flash is proprietary; Adobe is the only one that can release Flash for different operating systems).
There is also a website known as DSiTube (http://www.dsitube.webs.com),
which appears very similar to the unrelated site DSiCade. However, the owner of DSiCade denounces DSiTube, calling it a hoax. There is not adequate information to prove it either way, however; DSiTube currently does nothing and the owner promises that success will be seen shortly.
The best that can be done right now is to wait and see if a Flash Player will ever come out for the DSi, or a YouTube app similar to the one on the iPhone/iPod touch. I'm sorry if this disappoints you.
However, if you just want browser games that are compatible with the DSi (although they are playable in Firefox 3.5 and likely many others), DSiCade (http://www.dsicade.com/)
and DSOperaSDK (http://dsoperasdk.com/)
are possible options.
(no DSi listing)
(public denouncement of DSiTube)
(Flash card workaround for inability to run software from SD cards)
(statement that YouTube/Flash player don't work in "Specific website micro reviews" section)