I have to admit that I have not followed e-mail clients much over the years. However, I do not think that too many if any e-mail clients screen links, however any anti-spam and anti-virus software will. Generally, these e-mails are returned as an attachment in a text e-mail warning that there is a problem. Some anti-spam software will drop the e-mail entirely as an option. It may also be possible that a link in an e-mail is converted to text only but that should not be a big problem.
I am very sure that robots.txt is not a problem either. Robots.txt are for spiders so web browsers should not check robots.txt file as well as any e-mail client.
The first thing I would check is to make sure that your domain name and IP address are not on a blacklist. My favorite is http://mxtoolbox.com/blacklists.aspx. If your site is not on a blacklist, then I would try and find someone who is having trouble and have them forward the e-mail to you. From there, you can check the raw e-mail code for issues.
Another consideration is that some firewalls have anti-spam and anti-virus checking which may contain checks for phishing and certificates. It may be that firewalls are blocking access. In this case, the phishing checks often use blacklists so you may/should know this using the site above. If your links use HTTPS and you do not have a certificate, then I would either change the HTTPS to HTTP or get a certificate and check that someone can access your site using HTTPS.
Otherwise, I am out of ideas. If I think of anything else, I will update this answer.