This site has videos of all the major parts of the Okami game. I don't know if I'd call it "speed run" though, since they went through all the dialogue at normal speed rather than skipping it (which you can do starting with your second run).

This video shows a full runthrough of 11 hours, 44 minutes.

My own current record for a complete run through from start to finish is 6 hours, 14 minutes.

My run wasn't "perfect", so my guess is that a few minutes could be saved here and there, to maybe get the time to just under six hours, but I'd be really surprised to see anything faster than that, unless there's a way to further skip all the "loading" screens. If someone knows of a "fastest time" website for Okami, I'd be interested in seeing it!

Elonka :)