Where can I find plant fiber?

Solution 1:

Some trees (which may not be present on every planet) yield plant fibre when cut down. Just use your matter manipulator.

Vines work as plant fibre. You can find those growing on cave ceilings. Cut them with the matter manipulator, too.

You can grow some yourself, which is renewable, but takes some time and requires some preparation. You'll need a few seeds (you start with those) and basic stone tools.

Solution 2:

Wheat often drops Fiber, so farming wheat on a large scale could be a solution, assuming that you didn't manage to find one of the large flower-like trees (not available on all planets) that drop Fiber instead of Wood.