What is the ruling on working as a freelance web designer if the client pays money in advance to an interest-based bank account belonging to the intermediary website, and then the intermediary website sends the money to the designer's account via the interest-based PayPal service?

Working in website design and selling them is permissible in general. However, if the designer knows or has strong suspicion that the website will be used for something prohibited by Islamic law, then it is not permissible for him to design it, lest he be an accomplice to sin. But if he does not know the client's intention, or does not strongly suspect that it will be used for something unlawful, then there is no harm in designing it and receiving payment. The fact that the payment comes through an interest-based bank does not affect this.