Welcome to Android Central! If you entered these contacts into the local Phone account, rather than your Google account, and you didn't back them up anywhere, then you're stuck. There's no easy way to pull locally saved contacts, since they're not in a discrete file you can access -- they're saved to an SQLite database in a protected system directory. Your only option would be to get the screen repaired so you can access the phone again.
In the future, I strongly recommend entering your contacts into your Google account, so that they sync automatically to your Google Contacts in the cloud.