Is there any crawling plant that, when it's crawling among others of its species, will actually grow into one another and eventually share veins?
Unrelated: this is for a project to provide a living shade canopy over a suburban street.
Most trees have fungi that share nutrients with them. Trees can share nutrients through these underground networks of mycelia or can do so in a more direct manner by grafting branches together.
If you had a vine with a woody stem structure you could probably graft together different plants or they might naturally grow together in a process called inosculation which is often found when you pleach trees together.
I don't know of any crawling plant that you could use but many trees and some vines are potential candidates.