Welcome to the board! I am not familiar with White machines, but have used Singer buttonholers (that look very similar to the one pictured) with no problems. Are you sure the base plate is the correct one for your buttonholer? Sometimes vintage parts get mixed up in sewing baskets, etc. and it may not be the plate that goes with your attachment.
Sorry I can't be of more help, but I am sure someone else here will be able to help you. If not, there is a site for vintage machines called Victorian Sweatshop and someone there may be able to help. Good luck!