Could salt and vinegar kill blackberry canes?

Enough salt and vinegar will kill anything, but it wouldn't be easy. The best way to get rid of blackberry canes is to cut them all down, then dig up as much of the roots as you can. Wait for them to sprout again, and dig them up again, and repeat this process as many times as you have the time and patience for. Each time the plants re-sprout and you dig them up again they'll be weaker. Then you can "solarize" the soil to kill off any remaining rootlets. To do this, spread heavy, clear plastic over the affected area in the hottest part of the summer. Leave the plastic on for at least six weeks. The sun will heat up the soil enough to kill not only the plant rootlets, but most of the pathogens in the soil.