As per Quran, the actual ethical prohibition underlying all economic transactions is لَا تَأْكُلُوا أَمْوَالَكُمْ بَيْنَكُمْ بِالْبَاطِلِ:
يَاأَيُّهَا الَّذِينَ آمَنُوا لَا تَأْكُلُوا أَمْوَالَكُمْ بَيْنَكُمْ بِالْبَاطِلِ إِلَّا أَنْ تَكُونَ تِجَارَةً عَنْ تَرَاضٍ مِنْكُمْ
Believers! Do not devour one another’s wealth by wrongful means except through trading by mutual consent. (4:29)
This verse prohibits a person from devouring other people’s wealth through means which are against justice, honesty, fairness and against the good conventions of a society. Obtaining money through illegal gratification, theft, extortion, lying, co-operation with evil, embezzlement, misappropriation, consuming unclaimed items without publicizing them, all come under it. These evils require no further discussion since they are universally acknowledged sins in every society and in every religion. Transactions and activities which become a source of deceit or damage for the parties involved are also corollaries of this directive.
Any business activity which contains any such element in it, is to be considred prohibited. Other than that, all business activities are considered as allowed. Dropshipping doesn’t have any immoral element in it which should render it prohibited on the basis of its nature solely.