They do if they want to get them away from Groupon.
I have e-commerce customers who love Amazon Merchant because it's cheaper to acquire customers than SEO or buying mailing lists. But nothing pisses them off more than a customer who comes through Amazon a second time, costing them all their profit. They need to save the customer info in order to make the second sale a direct sale.
The same dynamic works with Groupon, except in person. If you don't follow up with them directly, there's a good chance they'll just wait for the next Groupon promotion, whether it's from you or your competitor. There's anecdotal evidence that this is already happening. A fresh coupon in your in box every day can get awfully addicting.
But for lots of retailers like salons, specialty shops, services, even restaurants, Groupon can be an effective way of building their customer book. When you call them back yourself, you have a much higher probability of converting them into a permanent customer.
One merchant I bought from had a direct store and an Amazon store. The Amazon store price was higher (by about $1.50 on a $20-ish item), but I still ordered from Amazon because I expected the merchant to take care of Amazon customers better wrt shipping time etc. :)
As a buyer, given a choice I would rather order from the Amazon store than directly from a no-name merchant.
I think your e-commerce customers just need to raise their Amazon store prices.
Those who sell commodities don't have that option. Every time any of them has tried, sales went into the toilet. You have to stay competitive when Amazon's web page shows your item right next to your competitor's.
The same reasoning for websites periodically sending their users emails is at work here. Remember how Facebook used to send email notifications that you've received a new Facebook message, but wouldn't tell you the content of that message in the email? "Hey, we have more goodies back here - come visit again."
My wife's favorite spa has also figured out that sending me follow-on ads is a pretty effective sales pitch, as I semi-routinely say 'hey, she deserves another half-day at the spa.' They got my email address from an initial online sale.
I don't get any follow-on advertising from donut places, but there's a local restaurant who uses email to fill out slow nights with some really nice prix fixe dinners. Every now and then one shows up in my inbox, sounds delicious, and I don't feel like cooking, and they make an additional sale. They started doing so after I made a reservation on OpenTable, and our dining frequency jumped substantially.