What is the ruling on accepting a fixed amount (e.g., $10) for each client brought to a company, knowing that this amount is not a percentage of the company's profit from the client, and there is no fixed monthly salary, but rather the income depends on the number of clients?

If the company pays a lump sum for bringing in customers, this is permissible as long as it does not involve assisting in the sale of something prohibited. This is called brokerage (samsarah). This transaction is not a lease (ijarah) because a lease requires the knowledge of the wage and the work, and the work here is unknown. Rather, it is a جعالة (ju'alah - a promise of reward for performing a specific task).