I'm thinking/hoping that my issue is that their host doesn't allow the mail through because it's not being sent by someone on their domain. I've read a bunch about that and it didn't really sink in until I just read this article: http://wiki.dreamhost.com/Sender_Domain_Policy_and_Spoofing
Their host is not dreamhost, but I imagine others have the same policy. So I've now tried it with a real email address from their domain in the from field and the user-input info as a reply to address in additional headers.
I won't find out until tomorrow if it worked or not, but it might be worth a try for you.
As I'm writing this, I'm in doubt it will solve my issue because their e-mail is hosted separately from their web hosting account so maybe I need to use the SMTP method mentioned at the bottom of the article.