Is it permissible for a young man to send a message to a girl he does not know – with vast geographical distances separating them – expressing his desire to marry her, and asking her to send her guardian's account information so he can formally propose to her?

It is permissible for you to send a modest letter proposing to the sister in question, either explicitly or implicitly, and to request her guardian's contact information. It is also permissible for a mature woman to receive a proposal directly for herself. Correspondence is a means to a legitimate end, which is the proposal. The proposal, generally, is a means to marriage, not a condition for its validity. Its ruling is permissibility according to the majority of scholars, and it is recommended by the Shafi'is.