No, it will not be a problem. Each patient will get a personalized and individual link in the email, each with different access keys and encryption keys. Obviously the emails will end up in the same inbox (Gmail, Yahoo, etc.) and that may trigger privacy concerns that would apply for any sort of clinical email to the shared email address.
It's generally a good idea to warn patients about the privacy implications of sharing email addresses in a email consent form that a patient must review prior to receiving clinical web questionnaires. There are some very well-thought out email consent forms in the Ocean clinical library and shared by other clients if you don't already have one.