Technically yes; practically no. You will be limited in the types of activity you can accomplish.
I do much of my daily banking and retirement activity online only. But I still have to keep a local credit union account because there are just too many times in a year when I need to visit a branch.
- Large cash deposits or cash withdrawals
- Bank services like money orders or notary public
- Screwy scenarios where you need a person to make a decision. (Being face to face is important.) My neighbor wrote me a check and was sloppy on the amount line. It was rightfully rejected by online deposit, but a cashier was able to verify me and make sense of the check were a computer couldn't.
All of my online banks can receive and send money electronically, but will fall back to paper. For example, my simple.com account (which is in beta) couldn't transfer funds between banks for free*, so I used their "electronic bill pay" to send myself a check to deposit into another account. They mailed me a paper check. When my ATM card didn't activate with them, I called and spoke to a person.
And when some situation comes up that you do need to speak with a human, even if that is only once every couple of years, then you will realize that online only just isn't a reality yet.
But it sounds like it would be a secondary bank for you. You can probably accomplish what you want. Cut up the ATM card they send and rely on bill pay, money transfers and electronic deposits. You will still have a back up.