What is the ruling of Islamic law on purchasing shares of modern transportation companies owned by non-Muslims, given that there is no usury in their transactions and their apparent business dealings are permissible, but it is unknown how they manage their financial affairs or whether they deposit their money in usurious banks or not?

You must ensure the existence of a real company with an official registration and whose activity is transportation, as a precaution against dealing with a fictitious company whose transactions involve usury of excess and usury of delay. If the company's existence is confirmed, then you must ensure the purity of its shares, meaning that it does not borrow or deposit with interest, because mixed or forbidden shares are not permissible to trade. This can be ascertained by examining the company's financial statements. It is not sufficient that they are Orthodox who prohibit usury, because dealing with usury is widespread among many of them.