Almost any financial institution has the technical ability to do this (simply called sweeps, auto sweeps, or deposit sweeps); the issue you face is finding an institution that is willing to do it for you. I think you will have the most luck at your primary financial institution where you currently keep the majority of your banking relationship. You will have better luck at small-town banks and credit unions. The mega banks will likely not waver from their established policies.
Deposit sweeps are common for business accounts. They are usually tied to a savings account, which is usually held within the same institution, however this is not a requirement. The sweep can send money to any US bank if you can provide the routing number and account number. The sweep will establish a peg balance, or floor balance, on the checking account. At the end of the day, any amount above the peg is swept into the savings account automatically.
I doubt you will find what you’re asking for within an online banking system. You will likely have to go into a branch and speak with a personal banker. Explain to them you want to establish a sweep on your checking account and want to send the funds to another financial institution. You will have better luck asking for a peg of $100, or some other small amount. They may not take your request seriously if you want to completely empty the checking account to zero.