Hi Everett,
In this case both the sites would be tied into each other and aren't that different, but my thought was that separating the online store would give us more flexibility with the root domain. If I implemented this, their WP site would be customersite.com and the e-commerce side of it would be at shop.customersite.com.
Their current website is through Square (not Squarespace), and it's a watered-down version of Weebly. Square also handles their online payments, in-store payments, customer loyalty system, and inventory management, so that's why we were thinking of relegating it to a subdomain instead of switching everything over to WordPress. Thankfully, Square makes it really easy to change the site address to a subdomain, so there isn't going to be a ton of migration work involved.