If you are referring to horticultural "vinegar" (acetic acid), that is not only very caustic and hazardous to use, but only a contact-kill treatment, meaning that only foliage directly sprayed will be damaged by the acid, leaving roots unharmed for regrowth. If you were instead referring to household vinegar (much weaker in concentration), that is too dilute to have much impact, even though it acts in the same way.

Removal of perennial weeds can be difficult even with the use of a systemic herbicide to kill roots (as it typically takes several repeat applications to get all the regrowth). If you're avoiding herbicides, the only alternative is to physically kill the plants by digging up what you can, as often as it reappears, until whatever root fragments that are left exhaust themselves and stop regrowing. Smothering the weeds by blocking their light for weeks at a time is less likely to work well, in part because Canada Thistle can easily creep out from under such a cover, and so much root energy stores remain for later growth that it will probably just wait-out any attempts to stifle growth.

We don't recommend the long-term use of weed barrier material (by "plastic," we presume you mean one of the woven materials that will still drain water through it). They can be useful for smothering weeds before removal and mulching/planting, but left in place, they might interfere with drainage and oxygen permeability (roots need to have enough oxygen in the soil to stay healthy), may shed micro/nanoplastic pollution, and could become entangled with any desirable plant's roots over the long term, if there are any shrubs or trees in the area whose roots reach that far out (or any veggies or herbs that are particularly deep-rooted, depending on how far down the barrier would be). It also won't be much of a barrier to tough spreading rhizomes and roots of weeds like Canada Thistle, which might eventually punch through the layer or just travel beyond its edges to keep sprouting.

If you plan on leveling/re-grading the soil in that area, it would probably be more useful to take that opportunity to dig out what weed roots/rhizomes you can, and then to vigilantly scout for any regrowth once everything is back in place, and either pulling/digging those growths up promptly or spot-treating them with systemic herbicide if you are willing. (This greatly reduces the herbicide volume needed.) We don't know exactly how far down you'd need to dig for this clearing, but you can do a test dig-out first to see how far some of the plants are rooted-in, if practical. Since our web page (linked above) notes that Canada Thistle is capable of rooting as far as 3 feet down, and Johnsongrass can spread as far as 200 feet, this is why the use of systemic herbicide is typically recommended, since the alternative is a much longer, drawn-out process of slowly starving the plants by preventing photosynthesis. The faster you can remove any regrowth after each removal effort, the faster the roots will starve: it uses-up stored root energy to regrow, and if the plant can't replace that and fuel growth by photosynthesizing for very long, then it will eventually exhaust itself. Any growth missed or left to photosynthesize too long is working against you by allowing for some recovery by the plant.

If you are willing to use even a limited amount of herbicide, try a systemic (glyphosate being the most likely ingredient that is effective) application targeted to regrowth after the initial one or two removal efforts take place. That way, there is much less growth to spray. Glyphosate binds to soil well and is resistant to leaching, compared to some other systemic alternatives (not all of which impact grassy weeds) that are more risky with regards to polluting groundwater or persisting in the soil in an active state.

Miri