How can I access a backup from my old phone in order to obtain the contacts?

 Backups - Locate backups of your iPhone, iPad, and iPod touch

Restoring from a backup erases all data currently on your device. Any data newer than when the backup was made will be erased. A restore is an all or nothing restore. You can't access a backup and only restore certain files.

Backup - Restore your device from an iCloud or iTunes backup

If you were syncing contacts with iCloud, have you gone to Settings/Your Name/iCloud, signed in with the same ID that you used on the old phone, and turned Contacts syncing on? Have you checked iCloud.com/Contacts using a computer browser to see if the contacts are there? If necessary, go to iCloud.com/Account Settings, look under Advanced at the lower left, and select Restore Contacts.