As a former employee of the Staples Copy Center, I’d probably have to guess they’ll say no. The laminating services are usually meant for flat sheets, and the cords on the scapular means it wouldn’t be flat. We used to deny laminating little hand-made bookmarks with 3D stickers, ribbons, and other embellishments for the same reason.
I’d recommend trying it yourself, or buying a pre-laminated one.