Applications can be done online. However, there seems to be no ACH transfer option for funding. The application only has a check or wire option for funding. You can designate payable-on-death ownership with multiple beneficiaries in the application.
BofI is one of the oldest online-only banks. It opened on July 4, 2000. I first reported on BofI last July. BofI has never had competitive online rates for its savings account or money market account. However, it does have some nice checking accounts, especially for Seniors. The rate is 3.65% APY for up to $25K. There are no minimum balance requirements or monthly fees. It offers free checks and ATM reimbursements of up to $8/month.
Bank of Internet USA is FDIC insured.