Since there are no replies to this problem, I am guessing the authors do not care about my issue which may be problematic to others. I say this because other issues that are more recent than mine are responded by the author. I might as well find a different plugin. Thanks for being officially unsupportive.
It seems like it’s a hosting issue: the form-generating file is being accessed directly, which some hosts could see as a malicious thing. Contact your host and make sure they’re allowing access to the /wp-content/plugins/constant-contact-api/form.php file.
Please note: this is a free plugin. If you want immediate support, I recommend you hire someone. WordPress has a very active development community.