Is it permissible for an ambulance worker to accept money given to him by patients or their families, without him requesting it, while he fears his intention might change in the future, and while his official employer rejects accepting money, yet implicitly allows it?

You did well by refusing to accept gifts from patients and their families, and by prohibiting the driver from doing so. Accepting and soliciting such gifts is forbidden in Islamic law (haram), and it is not permissible to take them because they are ill-gotten gain (suht). Neither the need for money nor the widespread practice of taking them makes them permissible. And whoever gives up something for the sake of Allah, Allah will compensate him with something better.