Is it permissible or forbidden for a brother to invest clients' money that was sent to him to purchase and complete orders, and to delay its payment to factories and companies?

If your brother acts as an agent for clients, he is not permitted to dispose of their funds except with their permission. He is liable for this money if he invests it without the owners' consent. As for the profit resulting from this investment, there is a difference of opinion among scholars. Some scholars hold that it belongs to your brother (the Shafi'is and Malikis), while others hold that it belongs to the owners of the money (the Hanafis and Hanbalis). However, Sheikh al-Islam Ibn Taymiyyah chose that the profit should be divided between the transgressor and the owner of the money, because it was generated by the money of one and the effort of the other.