Fixing Curling Vinyl Flooring
One option would be to install quarter round next to the baseboards. If the edge has barely started curling up, this would hold it down.

Otherwise, yes you will have to peel the vinyl back to apply adhesive to it. One method to make the floor a bit more pliable is to use a hairdryer to heat it up. Just make sure you don't overheat it too much. 

Before applying the new adhesive, you may have to scrape off some of the old adhesive, and clean off any resulting debris.

Here's a good article with more details: Link