Do we need any special permission from a customer to send an e-mail to them to say Thank you for applying for a loan and that we have started the loan process. We would not be sending them any documents by e-mail just a Thank you contact. If the customer gives us their e-mail address then does that give us the right to e-mail them about the process of their loan? Thanks.

If they are not on an opt-out list you may contact them in this manner. That doesn't mean they are expecting an email or to have to react to it such as you requesting more information to proceed.

This would be transaction related based on your description and not advertising other products. It shouldn't fall under the CAN SPAM rules, but again, it can depend on what the email contains and if the customer has opted-out in the past.

Since your proposed "thank you" email does not contain any federal disclosures that must be delivered "in writing", then you do not need the customer's consent before sending the message.

The You-CAN-SPAM Act allows you to send what you want unless the person has opted out. If it's easiest, you could abide by any prior opt-out instruction from each credit applicant (as Andy suggests), but I'd call this a transaction or relationship message & that exempts it from opt-out.