Not really. With https luckily being the default, at most they could get the sites you were going to (I don’t think dnss is dead, but it’s been very slow to grow unfortunately).
They could probably see if you’re checking Amazon or Google, but wouldn’t be able to see what you’re looking at exactly. Theoretically they could use cameras and or triangulation to see what you’re in front of when you use the Internet, but a VPN would still show traffic so they’d know you’re looking up something.
The big thing this would do is act like a loyalty card… They give you some amount of benefit in exchange for tracking your purchases in ever higher detail. Mostly it’s just like that, except they’d also be able to see how long you are in the store, and ideally they can link it to your purchases so they can infer more about it
FWIW, I wouldn’t only consider giving them a disposable email