We are currently a bank in organization. Does anybody know if there is any regulation that would restrict us from contacting investors/potential customers by email? We are trying to be as paperless as possible and send updates about our stock offering by email if an email address was provided. Do we have to provide an opt out option if they voluntarily gave us their email address?

Having worked with a few deNovos I am not aware of any regulations prohibiting your using emails to contact potential customers/investors. As with all emails, it is good to have an opt in/opt out option at the bottom.
Be careful of what you say in the email, regarding the offering. It might be best to just ask if they are interested...if so, please let you know.