She plans to talk her father into buying her a new car.

Literal

She [topic-は] father [object-を] persuade [then-て] new-car [object-を] buy-receive [intention-つもり] is.

説得して新車を買ってもらう chains persuasion (説得して) with the benefactive result (買ってもらう = 'have someone buy for me'). ~つもりです states her intention.