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.
CampaignMonitor provide this service! Mail Chimp from what I gather do not provide this. It requires you to use the form and you have to have their logo underneath.
I think the site I provide uses this service as well. It interacts with createsend.com, which I thought was the provider.
Glad that helped! Yes, I haven’t used MailChimp. so wasn’t sure how it worked. Anyhow, createsend.com is indeed a CampaignMonitor URL. CampaignMonitor is really nice to use, I must say. Have fun with it.