Yes vinegar does kill black mold and about 82 percent of mold species due to the vinegar being a mild acid that the mold cannot survive when it comes into contact with it.
To kill black mold and any other mold use undiluted white vinegar and put it in a spray bottle and spray the vinegar directly onto the mold and let it sit for an hour and then scrub the surface down.
The mold should then come off the surface and the mold should be killed using the vinegar.
You can also kill mold and black mold by using apple cider vinegar too but white vinegar works just as good at killing mold off of surfaces.
Bleach is another way to kill mold but the vinegar is less harmful to your health than bleach is and you don't have to breathe in the bleach fumes.
If the mold is on some drywall it's best to cut out the mold from the drywall and replace the moldy section of the drywall to completely get rid of the mold and then remedy the moisture problem that caused the black mold to happen.