Adding customers (not members) to the MailChimp list

Sure.  There are lefty of plugins that can integrate the MailChimp API into your site. Which one depends on your cart system.

Actually, after thinking about this more, it seems that what I want to do - add everyone who buys something to the MailChimp mailing list - is not really allowed.  You need someone's permission to add them to the list and just buying something doesn't give that permission. It seems that I need to add a MailChimp form to my site, where people ask to be added.

Not entirely true, you just need to enable double opt-in on the MailChimp side.  So the checkout form (with appropriate language, maybe even a checkbox)  serves as the mailchimp form and then they also have to click a link in an email from mailchimp to confirm. That's perfectly fine.

Ok.   So let's suppose I have a checkbox somewhere with the label "Add me to your mailing list."  Now how do I make submitting  the checkout form and finding that box checked trigger the transfer of the name/email to the MailChimp list?

Thanks for your help.
Steve