This is one of those things that simply must be done, just for the sake of the 'Coolness Factor'. Kind of like the tiger trap in disneys 'Swiss Family Robinson' (It's gotta have a tiger in the pit, because, well, that's the whole point!)
When I was a kid in southern california my grandma took me to a park where they had walkways covered with grape vines growing up both sides and over the top, making a layer a foot or so deep. After running around for a while in the walkways I climbed up on top and started playing around on top. I couldn't run easily up there because there were lots of small gaps, holes and weak spots (hence in the pictures posted here, the walkways are stone laying on the
roots). It did carry my weight easily, even though it was just an arch. I was up there for a long time (probably a half hour or more) before my grandma saw me and threatened to beat me within an inch of my life if I didn't get down immediately.
From that
experience, and a huge old grape arbor we had behind a house in west Texas, I think one could train grapevines to make a bridge pretty easily. I would want to run some vines up at an angle to form intermediate supports, like in a suspension bridge if the span was too big, but I don't see a problem if you took about 10 years before you tried foot traffic and laid down some kind of walk way. Maybe a woven willow mat as the walk way, maybe even rooted at the ends.
It would be even more cool if you intemixed your species, using
trees and vines. Maybe pollarded cottonwood or willow underneath as a support post, growing in the wet low spot, with the new growth from the cottonwood/willow trained outside, around the bridge.
Since a lot of trees, as mentioned by others, will grow together where they touch (especially if encouraged) you could use cross growing branches to make a kind of grid structure at the sides that would provide a lot more strength for the weight, although I'm not sure how long they have to grow together to gain sufficient strength, (once again, probably a ten year
project).