Is it permissible to create a Gmail account with false information and an unreal tax number via a VPN to purchase games at a lower price and sell the credit at a higher price, and is it permissible to do so using a friend's information with his authorization?

It is not permissible to use a VPN to input information that is contrary to reality (lying). If this results in a violation of the seller's condition, it is an infringement upon their right and an unlawful appropriation of their wealth. Similarly, the use of another person's data is subject to the condition of the producer or seller; if they prohibit it, then adherence to that prohibition is obligatory. This ruling applies if the VPN is used for permissible matters. However, if the game itself is forbidden or disliked, the matter is more severe.