Take full responsibility for the car accident and serious injury on the way to help the company deliver goods. Can I ask my boss for compensation?

Whoever is seriously injured will take full responsibility. You didn't make it clear.

If your eldest brother is seriously injured and hospitalized, it depends on whether there is social security, which can be reimbursed.

Auto insurance is for third parties, and it has nothing to do with drivers, unless you buy driver insurance.

This situation belongs to work-related injuries. If you have insurance, you must be insured. If there is no insurance, you can ask the boss for compensation (provided that you are responsible), but if it is your responsibility, the boss may not be willing to pay compensation. If it is the other party's responsibility, you can ask the other party and the other party's insurance company for compensation.