I grew a vining Korean summer squash from Kitazawa last year: Teot Bat Put. It's a hybrid roundish (aka avocado squash) C. moschata.
https://www.kitazawaseed.com/seeds_summer_squash.html
Franchi Sementi has a zucchini called Zucchino Alberello di Sarzana ("little tree of Sarzana"). It's supposed to be a vine but I haven't grown it.
http://www.growitalian.com/zucchini-...lo-of-sarzana/
Another alternative is to plant a vining winter squash and eat it in the immature stage (as a summer squash). Delicata has the "right" shape, and butternuts are close.