I'd like to invite a new member; however, I'm receiving an error: "You can't invite users directly, but we'll forward your request to the owner or manager for approval."

If the Organization Setting "Let Members Invite Others" is toggled off, then only owners and managers can invite new members to the organization, as covered in the Permissions section of the Organization and User Management docs here.

Please reach out to your org's Manager or owner listed on the Members page for assistance.