I've used vinegar and although it took awhile, I eliminated the nutsedge and everything else in my parkway. I wanted to plant a mix of native grass seeds. But then the city came along, flooded it and did this or that so weeds of all sorts grew back. Didn't have time or energy (or trust that same thing wouldn't happen again) to do over. However, IF you plan to use vinegar there are a couple things you should know.
Someone already mentioned that it's not the vinegar that you buy in the grocery store. You can buy a stronger vinegar specifically designed for weeds at your local garden center or big box store. The original recommendation was to use '20% Vinegar'. Later, the garden guru changed that to a lower %. I have some in my garage and can edit this later with the specifics about proportions. It's just that my garage is a scary place due to unknown critters in there.
And β this is very important β you have to do it on a very sunny day. And since it's a spray, you'll want to spray as close to the bottom of the weed as possible. This will not totally eliminate most weeds. After the tops die, you'll still need to do some digging or hand-pulling.
Oh, one more thing that is MOST important. Wear gloves. Wear old clothes. Shoes and heavy socks. Totally protect your skin. Even your hair. We are talking about harsh stuff. If any splashes on you, wash it off immediately!
EDITED: I braved the garage to read the label on the vinegar. Not only does it need to be a sunny day, but it has to be at least 65F degrees. It does mention to be aware of drift. And don't use on edibles (duh) and so much fine print that I didn't want to hunt down my magnifying glass. In fact, next time my city has a free chemical waste disposal day, I know what I'll be doing with the vinegar.