My bank (Capital One 360) is also an online only bank, but will send you a notarized verification of deposit statement. I've used this twice now to apply for visas in Category F, without any problems. I printed out the monthly detailed statements then used the VOF as my original supporting document.
The only problem I had was making sure I got a "notarized" VOF. The last visa application was stressful because the bank sent a regular printout without a signature at first and I had to request it a second time which made everything get too close to my deadline for comfort. I learned to stress the word "notarized".
Oh, and they did fuss a bit at first about sending this to me but eventually came through when I pointed out I opened the account in the first place because they offered this service.
