To unlock the pickup truck you need 20 bronze trophies, 10 silver trophies, and 5 gold trophies.
Every vehicle related achievement requires a different amount of a certain type of trophy. To view these requirements you can find signs posted in and around the headquarters that show how many bronze, silver, and gold trophies you need to unlock. The unlock requirement for the pickup truck can be seen in the garage of the headquarters.
To earn trophies you have to complete deliveries. There are some deliveries that need to be completed quickly and there are some that need to be completed with minimal damage to the package. For me it is easier to earn gold trophies on the deliveries that don't require speed, you can take your time and deliver the package safely.
When you earn a trophy, you also earn trophies of the levels below. So if you earn a gold trophy on the first delivery you also win a silver trophy and a bronze trophy. You can only earn each trophy once. You can play a delivery over and over in order to improve your trophy, but if you already have gold it won't count again.