If you have stabilized vinegar, vinegar that's been strained and heated, like bought white vinegar, it doesn't need to be refrigerated.
If you have an actively fermenting vinegar that's reached a stage of a little sweet and a little vinegar, it's a very subtle balance, you could slow the fermentation by refrigerating it. It won't be as acidic as a fully fermented vinegar.
I just finished up 2-3.5 gallon batches of nectarine vinegar, it has a slight sweetness and an active mother. (like the mother you would find in natural apple cider vinegar, or the mother like scoby you might find in kumbacha) You can let it continue to ferment or slow it down with refrigeration.
Infused vinegars, like say, fig infused blended AC vinegar, may start to ferment a little further if left at room temperature because you've added sugar to it.
I'm often in the process of making vinegar, so I usually have some that is half sweet half acidic and I leave most of my vinegars on the counter, or stored in glass in the basement where it is cooler.
This is my current collection.