Is it a condition for a transaction to be considered an agency (wakala) that the purchased item be acquired with the principal's money and not the agent's?

If you are appointed as an agent to buy a phone, it is not permissible for you to buy it and then sell it to the one who appointed you unless you inform him, because an agent cannot sell to himself. Similarly, it is not permissible for you to add anything to the price of the phone if you bought it with your own money, because this would be an increase on a loan. However, if he appoints you and gives you the money, then it is solely an agency, and any discount or gift received belongs to the principal. It is not permissible to take any money except with the principal's knowledge, either as a fee for the agency, or by selling the phone to him after you have acquired ownership of it, or by selling it on a deferred payment basis. The principal must be aware that you are the seller.