Espadrilles are made with only vegetable fibers
Espadrilles (Spanish: alpargatas; Catalan: espardenya; Basque: espartinak), are casual, flat, but sometimes high-heeled shoes. They usually have a canvas or cotton fabric upper and a flexible sole made of esparto rope. The esparto rope sole is the defining characteristic of an espadrille; the uppers vary widely in style.
Esparto, halfah grass, or esparto grass is a fiber produced from two species of perennial grasses of north Africa and southern Europe. It is used for crafts, such as cords, basketry, and espadrilles. Stipa tenacissima and Lygeum spartum are the species used to produce esparto.
Though in the past they were peasants shoes, your people can use a similar approach to manufacture their shoes.
They might use vegetable fats to waterproof them.
Another approach is what Japanese do with their geta: wood and rope.
Geta (下駄) are a form of traditional Japanese footwear that resemble clogs and flip-flops. They are a kind of sandal with an elevated wooden base held onto the foot with a fabric thong to keep the foot well above the ground.