Drupal - Should I ignore the vendor directory in git for Drupal 8?

There isn't really a right or wrong answer, it will come down to your situation. Composer themselves have pretty strong feelings that yes, you should ignore it, and in an ideal world that's definitely what you should do, but it's not always that straightforward.

It depends mainly on your hosting and deployment process. If, for example, you're on Pantheon, and don't wish to implement a complicated upstream workflow, involving a dialogue with their technical staff, you're forced to commit all dependencies to the repo. There's no Composer sitting at the other end waiting to run when your code is pushed, if you don't push those files up yourself, they're not there, and things will break.

Similarly if you're on shared hosting and don't have a CI workflow, you're probably going to be forced into lugging all the files around with you in one way or another.

Basically if it fits into your development/deployment workflow to be able to exclude those files, it's a good idea to (for the reasons listed in the link above).

If it doesn't, committing them to the repo is absolutely fine.

Drupal core and modules update so frequently, often with security fixes, that it has become common to use composer to manage dependencies. Composer will place dependencies in the vendor directory, and then you don't need to commit them to your codebase. You can run "composer install" on the server and it will download the dependencies on that server. This allows your code repository to be lighter. If you are using this process, you wouldn't want to commit the files, so yes, you'd want to gitignore that directory.

If you want to get up and running with using composer to manage a drupal site, you can check out the drupal scaffold project.