In alignment with AWS documentation, we advise creating a new Backup Vault to store the Veeam Backup for AWS EFS Recovery Points, because a new Backup Vault doesn't have a default Access Policy.
Reference: Veeam Backup for AWS - EFS Backup IAM Role Permissions
As indicated in the Amazon AWS documentation:
Your account will always have a default backup vault. If you require different encryption keys or access policies for different groups of backups, you can create multiple backup vaults.
Furthermore, the Create Amazon EFS automatic backups documentation states:
The Amazon EFS automatic backup vault aws/efs/automatic-backup-vault is reserved for those automatic backups only. If you use it as a destination for other backup plans, you will receive an "insufficient privileges" error.