Should I mention I had no funding in my Ph.D experience?
I can't see into the mind of the PI, but think that you would harm your chances by suggesting that you might not need funding when they may be, explicitly, looking for funding sources. (Caveat below)

Since they ask for your proposed funding sponsors, you probably need to provide that to be in contention. If you can do that then an "Oh by the way, I've done research without funding" may not hurt you, but otherwise it could (possibly). 

But you would probably be wise to look elsewhere for a PI that can themselves provide funding for their postdocs. If they can't do that, I wonder how much commitment they will have to you. 

Of course, since the link is a very general one, they may not require all elements, especially self funding. You could also inquire directly about that.