You get the promotion event through the event section, when you interact with Wilfred looking for contracts. It's also necessary to max out your character first, otherwise the promotion opportunity will not arise.
Viewing post in The DeLuca Family 0.08.0.3 comments
You have to basically keep doing the missions over and over again until you reach level 5 (max level) to get the soldato promotion event. The missions are where Wilfred stands outside the office to Mrs. De Luca, and pick the contract option. Then, you will be given a few tabs including: a list of missions, equipment store, current inventory, etc. Wilfred is there during the morning til evening periods of the day.