The way that vinegar removes splinters from your skin is the vinegar shrinks the skin around the splinter and allows the splinter to come out easier.
You still need to remove the splinter even using vinegar but it makes the splinter come out much easier when you use vinegar.
To use vinegar to remove the splinter you soak the skin in the vinegar for a few minutes or even 10 minutes until the skin shrinks.
Then take your tweezers and remove the splinter and it should come out much easier.
The vinegar is acidic which is what helps your skin shrink and allows the easier removal of the splinter.
I've soaked my finger in vinegar before as well as my toddler sons finger in vinegar when he and I got splinters and it made the splinter come out easier.