Hi Jen,
Great question. I’m not a Constant Contact user myself so I don’t have any personal recommendation there. Does Constant Contact have a way to create a sign up form inside of your account and then copy the code for it and paste it into your site? That’s something that other newsletter services offer that might fit the bill here. You could either add it to a text widget or to the content of a page if it fits better. If you go the page route, click the “text” link at the top of the editor to switch from the visual editor to the text (HTML) one. That way you can add the code without WordPress mangling it for you.
Take care,
Bill