I don't see any reasons why a silicon valley startup couldn't. Lots of small bank startups sprout up all over the country. Also, Simple.com was a startup (they are now part of a very large bank) that got FDIC insurance through a partner bank.
Private corporations already underwrite bonds (and related debt products like pensions, corporate loans, annuities or mortgages). A deposit in a bank is not completely different from a customer loaning money to the banking institution (at low or zero interest) in exchange for on-demand funds availability.