Most insurance companies' normal private passenger auto policies do not provide coverage if the covered vehicle is being used to deliver property, i.e. pizza or people for a fee (salary, tips, etc). Coverage can thus be denied if an accident occurs while the personal vehicle is being used to deliver items such as pizza.
For example the South Dakota Department of Insurance states that most personal auto policies will exclude coverage for the vehicle while it is being used for "business purposes" such as delivering pizza. Many policies in fact have exclusion for using your car for delivery purposes such as this.
You can discuss with your current insurance provider if your personal auto insurance policy will cover you as a pizza delivery driver, but likely it will not. The restaurant you work for may already have liability protection for this type of activity to cover the restaurant's exposure. However, this may not protect you or you if an accident occurs. Typically, the restaurant will require additional insurance to protect themselves if you are negligent.
There are some states that might require the insurance company to allow the state's minimum insurance requirements to cover a private vehicle whether it is being used for personal or business use. So you can check with your state's insurance regulatory body to see if your state has such laws.
If state laws do not require your insurance company to cover the incident then you will likely need to get a commercial or business policy on his vehicle if you want to deliver pizzas and wants have your vehicle covered by insurance.
The reason for the business use or extra insurance coverage that insurers will require for pizza delivery drivers deals with the risks factors associated of this type of delivery job. A pizza delivery driver takes their car to a lot of places at different times of the day and night so it make is difficult for an insurer to accurately rate the risk of the driver and car.
Typically, no personal auto policy will cover "delivery" jobs. CarInsurance.com does not currently have a carrier that will cover this under a personal auto policy. You will need to purchase a commercial auto policy normally for which you can click here to get a quote.