After taking a look at your newsletter sign up page, I found the problem to be related to code for your search box. What is happening is that you have an HTML <form> element which wraps around the majority of your website to enable the search function. However, that same HTML form is catching the email signup form and instead of directing the signup to Constant Contact, the signup is being directed back to the same page on your website.
In order for the Constant Contact form to work, it needs to have a <form> element with the action set to "http://visitor.r20.constantcontact.com/d.jsp". One additional possibility would be to generate a button for people to click on instead of a form to fill out. If that is of interest, you can log in to your Constant Contact account, then click here. Next you will want to click on "Get a link to my website signup form" and create a button.
If you have any questions, please let me know and I will be happy to assist!
API Support Specialist