Hi bowlofnoodle . Welcome to the forums.
Yes, you can do exactly that with MailChimp, and with other services like CampaignMonitor (which I use). Once you've set up a mailing list for a site, you ask the system to generat form code for you. With CampaignMonitor at least, you choose what fields you want, so in your case, you just choose the default name field and nothing else. and the necessary form code is generated for you. Then you just paste it into your page, style to suit, and Bob's your uncle.