Are there are any states which offer the driver's license written test online?

No, but several, including Calfiornia offer sample tests online.

Not at home but North Carolina and most other states in the United States use a Computer Based testing system at the Driver License offices.

Not likely. The DMV wants to identify the person taking the test. That’s more difficult to confirm online.

And, more importantly, couldn’t you just look up answers in a different web browser tab or in the state-issued educational handbook?