If flight does not need to be sudden or often, the method that spiderlings use to fly may be plausible for a plant. Simply producing silk (or a similar strong, lightweight liquid solidified by tension) will allow a thread to build up pulled by the wind, until it is long enough that the wind lifts the plant into the air. A number of such silk glands at different positions on the plant would reduce the required length (and hence the required open space for lift off).
This approach requires only that the plant be able to produce a suitable liquid for thread formation. The rate of production can be low, with the thread slowly extending over a period of days or weeks before sufficient lift is achieved.
Although the wind provides the required tension once the thread gets going, a small amount of tension is required at the start in order to get the thread going. Spiders can use their legs to pull the thread out until it is long enough to release to the wind, but moving parts are very expensive for a plant. A less resource-intensive approach would be to start the thread with a weight to pull it downwards.
When the silk gland is mature, the tip of it is dislodged by the surrounding tissue drying out, and as it falls it pulls the silk liquid taut, turning it into solid thread. The weight is heavy enough to pull out new thread, but light enough to be caught by the wind to continue the thread growth rather than reaching the ground and stopping. The shape of the weight can be wide and flat and possibly feathered, to make it catch the wind despite being heavy.
Once the thread is long enough to reach further up from the ground where the wind is stronger, the strength of the wind breaks off the weight and the thread flies without further need for it. This keeps the thread light and reduces the length required for lift off.
Since this arrangement only gives rise to flight when complete, another reason is required for the evolution of silk glands and feathered weights.
For example, the silk may have first developed as a sticky liquid to trap insects (either as prey or simply to kill pests and parasites). Alternatively it could have been used to attach seeds to passing animals that brush against the plant. Over time the silk glands increased in size and allowed longer lengths of thread to be produced. The selective pressure causing this may have been that seeds hanging from longer threads attached to passing animals were more likely to get caught on vegetation and removed above ground, rather than in the animal's cave or burrow where there is no light. Being removed earlier would also be an advantage if some animals are inclined to eat the seeds during grooming. If the threads were used to trap pests and parasites, a selective pressure may have been that a longer thread meant that the struggling insects moved around more, attracting natural predators that would help keep the plant free of pests. Either way, the threads produced eventually become long enough to catch the wind, and then selection can act on that.
The feathering of the weight to allow catching the wind could then feasibly develop without alternative explanation, but it may already have started in the form of mimicking the insect pests in order to lure them to the silk glands. This would mean that no insect is required in order to start the thread. In evolutionary history it may have been insects that dislodged the weight and triggered the thread production, but eventually the weight could be released independently, giving finer control over the timing (and making it more likely to happen during high winds).
As an intermediary stage, they may be plants that form threads not long enough for flight, but long enough to entangle in animal fur/feathers so the whole plant can hitch a ride that way (probably only for short distances while the disgruntled animal makes efforts to dislodge it). Over many generations the thread becomes long enough to catch the wind and preclude the need for an animal.
Size and weight
Spiderlings are very small. Similarly, silk flying plants would be limited in size. In an Earth-like setting, these would likely be tiny - perhaps like mosses that do not need permanent roots and can colonise places beyond reach of most other plants. If you wanted larger plants (perhaps "kite trees"), then you would need a much denser atmosphere.
If you imagine a branching evolutionary tree, then even an Earth-like setting could have larger silk plants, but only the smaller species would be capable of flight. So then you could have kite trees with long threads trailing off into the sky, serving some other purpose like those described above. Their tiny relatives would produce far fewer strands but would take flight whenever the wind picked up, unlike their huge deep rooted cousins.