Is it permissible to give money to one's sister to buy a mobile phone, knowing that it will mostly be used for impermissible matters, and would the financier be sinful for that, especially with the fear of angering the mother and sister if the request is refused?

If you predominantly believe that your sister uses her mobile phone for unlawful purposes, then it is not permissible for you to help her purchase one, for fear that you might incur sin. You can advise her and explain the reason for your refusal to help her. If she promises to repent and you believe her sincerity, then there is no harm in helping her. However, if she insists on using it for unlawful purposes, do not assist her. Instead, explain the reason to her, and you can help her by purchasing another permissible item that she needs.