What you need to do is put a comment at the top and bottom of the cart and checkout pages (and contact form page) for customers to please add your email address to their whitelist, approved senders or address book.
Also put this comment on the Thank You or order confirmation page.
Make it obvious and note that they need to do this so that they can receive their order acknowledgement, shipping confirmation and communications from you about their order.
Adding a comment that if they do not receive your order confirmation to look in their junk/trash folders and then to be sure to add you.
This is something every eCommerce site just has to do to get their customer's attention and get white-listed so emails do not land in the trash.