According to a quote on the physics forum "Plants are usually charged negatively and emit weak electric fields." Reference However this seems to contradict the fact that plants are grounded and any static they build up would just be discharged into the ground.
Plants are not perfect conductors. Indeed mature plant material (wood) is a pretty good insulator. Therefore I see no reason to assume that (small) charges would immediately dissipate.
There are several reasons that charge could accumulate on the plant. Ionic transport within the plant is one. Ionic removal by wind is another. I don't know if either is actually present, but I don't see any reason to assume that the charge must be zero on every portion of the plant.
The Earth is negatively charged. As a result anything electrically connected to Earth sharges this negative charge. That includes plants, but also you and I should we be walking barefoot.