You won't see much plant-based armour in historical records, the already mentioned paper armour notwithstanding. As best as I can tell, the primary reason for this is that leather tends to be superior to most plant materials you can find. It's easier to process and form, and easy to combine with other materials to form composite armour.
That is not to say plant-based components were not used; besides the obvious use of plant fiber for textiles to make brigantine and other similar armours out of, for a long time, resin was the basis for most glues and laquers used in armourcrafting.
Japanese armourers would use resin-based laquer on metal or leather scales, improving their durability and - in conjunction with silken cord also used in copious quantities to hold the scales together - were able to fashion a composite armour of sorts, that was surprisingly resilient to piercing, given the low-tech materials used.
Romans would combine wood and glue to make plywood, out of which they would fashion scuta, which were used by the legions to great effect.
Options for plant-based armour
Assuming that plant-based components or strengthened textile armours are not good enough for you, ther are a few pre-requisites for purely plant-based armour to appear in the first place:
- Armour-quality/sheet metal is either too expensive or unavailable
- Leather is either unavailable or there is a strong cultural taboo against using it
- There still are metal-based weapons, or weapons of similar performance characteristics, necessitating use of armour
In that case, you can either take the cheap option, or get more creative.
The cheap option is that you posit the existence of a plant, maybe some kind of reed or something related to sugar cane, that has similar characteristics as leather and can be unrolled into sheets which you can process much like you would leather.
If you have to get creative, then processing is going to make or break this approach. @fredsbend already mentioned a couple of options involving selective breeding and exotic processing. One idea that I particularly like is "feeding" the plant metal (or some other) compounds so that they get incorporated into the fibers; this would let you create the equivalent of modern composite materials with relatively primitive means.
But in the end, if you were to create purely plant-based armour, your best bet is probably plywood. Even if you only have wood (and peoples like Pacific Islanders were able to fashion quite impressive weapons out of hardwoods), you can finely slice it to get veneer, which you can soak in water to make it more malleable and then layer on a form to create your armour pieces, rotating each layer to make sure there is no single "grain" direction that would make the piece prone to cracking.
For glue, you would use some kind of resin, and some other similar solution for the lacquer. Hopefully the result will be both resillient and flexible; the latter property will be more important (and likely provided by the resin more than the base material itself), as it will keep pieces from breaking of when hit hard.
If you have an exotic plant like the kind I described earlier, you would probably use it in much the same way.
Depending on the exact mechanical properties of the armour, you can either fashion an entire cuirass out of it (if it is sturdy enough), or use it to create segments for a lamellar armour, or scales for, well, scale armour. The nice thing about making an armour this way is that it's pretty easy to shape and you don't need a furnace or a blacksmithing workshop to make it work.
The end result will be a lot more laborious than leather armour would be (although it might end up being cheaper), and not as good as quality steel armour (or armour made out of fantasy metals), but hey, it beats getting skewered.