What is the relation between the doctrine of Five Vehicles (Pañcayāna) and renunciation of the world (pabbajjā)?
五乘教法与出家有何关系？

The Human Vehicle and Heavenly Vehicle do not require leaving home. The Pacceka Buddhas (self-enlightened ones), living in the absence of the Buddha Dhamma, led lives of hermits in the mountains and forests, but went through no ceremony of renunciation. As for Budhisattas, some lived at home, and some did not. In their case, it is not necessary to leave home. Whether or not they leave home depends upon the conditions (hetu-paccaya) of their maintaining the Buddha Dhamma and benefiting all living beings. For example, at the time of Sākyamuni Buddha, both Bodhisattas Mañjuśrī and Maitreya (P. Metteyya) were Bhikkhus, whereas Vimalakīrti, though a great Bodhisatta, was a layman (a scholar studying Buddhism at home). So it is only in sāvakayāna those who seek to achieve Nibbāna by becoming Arahats need to leave home and lead a homeless life.
人乘、天乘不要求出家；独觉生在没有佛法之世，虽是山林隐逸而没有出家的仪式；菩萨包括出家的和在家的人，不一定要出家，看住持佛法的因缘和利益众生的因缘可以出家，可以在家。如佛陀时代，文殊师利菩萨和弥勒菩萨都是出家的比丘，维摩诘也是大菩萨，则是居士（在家学佛之士）。只是声闻乘中为了证得现法涅槃，成阿罗汉的目的，要求修行者出家。