The regulation states the following:

Notice of incompleteness. If additional information is needed from an applicant, the creditor shall send a written notice to the applicant specifying the information needed, designating a reasonable period of time for the applicant to provide the information, and informing the applicant that failure to provide the information requested will result in no further consideration being given to the application. The creditor shall have no further obligation under this section if the applicant fails to respond within the designated time period. If the applicant supplies the requested information within the designated time period, the creditor shall take action on the application and notify the applicant in accordance with paragraph (a) of this section.

I can't find that "written notice" is defined anywhere. Is it acceptable for the written notice to be provided by email? I'm assuming if that is an acceptable delivery method we would have to make sure the customer has agreed to receiving communication via email and also verified that emails are being received?

Yes, email messages can satisfy requirements that notices, disclosures, etc. are "written" or "in writing"...BUT only after you have obtained the consumer's informed demonstrable consent in the ESIGN-prescribed manner.

I can't find that "written notice" is defined anywhere. Is it acceptable for the written notice to be provided by email?

See 1002.4(d).