I am answering two questions at once here about rose of sharon, simply because rose of sharon propagation by seed and preventing rose of sharon seedlings from popping up all over are two sides of the same coin, to a degree. It's just a matter of what your perspective is.
First, let's look at the issue from the perspective of someone who wishes to get rid of "volunteer" rose of sharon seedlings. For those of you not familiar with rose of sharon, it puts out lots of seeds in fall, and the seeds sprout readily next spring. The result is often unwanted rose of sharon seedlings growing all over one's lawn or planting bed.
How is this problem solved? Well, it's much easier to solve it through preventive measures, rather than tackling the problem after the fact. When the flowers of your rose of sharon are done blooming, simply deadhead them. This will nip seed production in the bud and eliminate all those rose of sharon seedlings. Make sure you remove not just the flower, but the developing seed pod at its base. The photo on this page shows what a rose of sharon seed pod looks like -- this is what you need to remove.
If you fail to implement this preventive control measure, then you will have a tougher row to hoe. Some readers ask me about spraying the young rose of sharon volunteers with an herbicide. I suppose you could “paint” an application of something like Ortho’s woody plant herbicide onto the little rose of sharon plants. I use the term, “paint” metaphorically to emphasize that you don’t want any errant spray landing on the parent rose of sharon shrub itself or on any other good plants, since the herbicide would harm them. Note, however, that such a method is not really easier than that old standby, hand pulling, which is what I recommend.
The only other methods I could mention are smothering the new rose of sharon plants with a tarp or preventing them from germinating by using a pre-emergent herbicide, as you would for crabgrass, in the spring. But tarps are not an option if the germinating activity is happening on a lawn (the tarp would smother the lawn, too). The smothering method might be viable in a sparsely planted planting bed. As for the use of pre-emergent herbicides, yes, they might work, but it makes so much more sense just to deadhead in the fall, doesn't it? And remember that, when you use pre-emergent herbicides, all seeds fail to germinate, not just the ones causing a problem.
Now let's switch gears and look at the issue from the perspective of someone wondering how to propagate rose of sharon by seed. Based on the foregoing, the answer here should be obvious: let nature do the work for you! Rose of sharon is easy to grow from seed, so let this fact work to your advantage. Just let the seeds drop on the ground in fall and winter, of their own accord, and wait for them to germinate in spring. Then dig up your new rose of sharon plants and transplant them to a new location.