How many times can same vinegar (after soaking onions in it) be reused again for soaking the fresh onions?
Do I need to take any special care w.r.t Vinegar for maintaining its freshness?
From a safety point of view, assuming you are keeping the vinegar in the refrigerator while soaking...
I would make the conservative assumption that the water and nutrients from the onions is diluting the acidity of the vinegar, and making it a more hospital environment for pathogens. Therefore, I would think of the vinegar as a fully perishable item, with a lifetime in the refrigerator of four or five days.
How many batches of onions could you soak in that time?
Of course, from the efficacy point of view, each batch of onions will flavor the vinegar more, and dilute it more--so at some point, you won't get the results you are looking for, whether that is a basic "velveting" of the onions from a quick soak, or a more pickling like change from a longer soak. You will have to decide when the results are no longer good enough.
The thing is--vinegar is very inexpensive, at least in the US. Is it worth reusing it, when you will get less consistent results?