You can increase your Wagon capacity 2 different ways: by levelling it up or buying special horses.
The game has 3 kinds of these animals that vary in terms of price and maximum load. You can buy new horses in the store, or by clicking on the animal next to your wagon.
Note that each horse can only complete a limited number of trips. Once this amount is reached, the wagon resets back to default.
Warning: trips aren’t saved if you switch to another horse before reaching this limit.