A non-owners insurance policy provides liability coverage for drivers who don't own a car. However, if the vehicles aren't insured and registered by your mother-in-law, you are still not going to be legal on Colorado roads.
Car Insurance Costs
From what I understand, using Amazon's in-car delivery should not affect your insurance rates. Because the couriers don't drive your vehicle(s) and are simply dropping items off in a parked car, I don't see any concerning factors an insurance company would worry about — unless a courier vandalizes or steals your car, which seems unlikely.
When you're delivering food, you're effectively using your vehicle for a commercial purpose. Most policies do not cover you to do a side job, such as rideshare or food delivery, without a special endorsement.
You will need to get a commercial policy since you will be a delivery driver. If you were going to drive for a ride share company you could get a personal auto policy with an endorsement to drive for ride-share.
In most cases, making deliveries in your personal vehicle would be considered a commercial risk. Even if it is only for a short period, you may not be covered if an accident occurs.