An alternate solution would be a sort of slingshot, using retractable tethers to temporarily latch on to celestial bodies of greater size than that of the plant creature itself.
Then, using the rotation of the celestial body along with a speedy retraction of the tethers, velocity can be achieved before releasing its hold. The leaf based solar sails, mentioned earlier, can take over further acceleration and steering after that.
Assuming the problem of survival in a hard vacuum is already solved, a plant based intelligent organism, capable of a high level of advanced mathematics, could be able to calculate trajectories to pinball between celestial bodies of appropriate size until the main destination has been reached.
We are definitely not talking about great velocities here. But then again, I hear the concept of time is relative to the size and metabolism of the perceiver, and a plant would most probably be a patient traveller.
Then comes the problem of calculating trajectories. The organism would need a naturally evolved sight or other means of scanning and interpreting deep space objects, while doing advanced astrophysics and celestial mechanics calculations, mapping its optimum path, before beginning its journey.
A question that follows would be how the hell the plant ended up in space to begin with, which also applies to the original question.
Did a behemoth plant grow into the form of a launcher for a sporelike chrysalis, towering several miles high, before shooting out that first tether, latching on to a passing asteroid?
Or maybe a single incredibly massive organism spread its burrowing roots throughout a planet to such a degree as to actually cracking the planet into multiple pieces, each one a newly born space traveling vessel.
Oh, by the way.
Rotation would be achieved by adjusting the angle of the leaves on one side of its frame to lessen the reflection of photons, like a rudder, using the same basic function as the solar sail.
It's not the most elegant solution, I know. Just my two cents.