Is it permissible to participate in a competition that requires the participant to photograph themselves with an advertisement for a poultry company in stores and post the picture on Instagram, where the winner is chosen randomly without requiring a purchase?

This work falls under the category of ju'alah (reward/stipend for a specific task) and not a competition. The basic principle is that the worker is entitled to the ju'al (stipend/reward) upon completing the work. Introducing a lottery here vitiates the ju'alah contract due to the presence of gharar (excessive uncertainty/risk), which is prohibited. The worker might or might not receive the ju'al, thus rendering his effort wasted. The gharar that corrupts transactions has conditions, including that it must be present in a financial exchange contract, be substantial, be in the primary subject of the contract, and that there is no recognized Sharia-compliant need for the contract containing gharar.