World wide all sorts of plants can be used. From stripped down fibres to bark. Have a google for Bangalow, Australian.
Those are water carrying baskets made by the native peoples from folded bark.
Here you can do it from rushes. The lake rush, used to be scirpus lacustris, but it's now schoenoplectus something or other.
When the rushes have been gathered and dried, and then re-soaked, they're like weaving with soft leather. We squeeze along their length to remove air before we weave with them. If you weave them tightly, twined weave, and weave them snug around a form (mostly we used the bottom of round flat based tubs now, but wooden formers were used in the past too) they shrink as they dry and they do become water tight, for a kind of damp value of tight. Mine holds a couple of litres. It feels damp to the touch, but I can carry water in it.
A wooden bucket works because the liquid swells the wood just enough to stop it leaking from between the planks while the binding straps stop them splaying out.
Basically all you need is something organic that swells with water enough to stop it leaking. Baskets do 'sweat' a bit though.
Thinking on it, the little foraging basket I made from the iris leaves, it holds water too, and I sewed it together with flax thread I made from flax I grew. The iris leaves I made into rope by hand so I can claim the whole thing as mine