So what, in the view of such a Buddhist, would be an example of something that is wholly reliant upon and determined by its own inner being? The universe? Consciousness?
That was a continuing subject of controversy among Buddhist philosophers.
A very influential and widespread early philosophical movement called Abhidharma tried to reduce all possible experience into what amounted to phenomenal atoms, called dharmas. These were elements of experience that weren't reducible to anything simpler. Different schools produced different lists of dharmas, usually amounting to several hundred. These were divided into conditioned dharmas, those that come into existence momentarily through causation before immediately causing another dharmas, and unconditioned dharmas, those that are eternal and uncaused. The lists of unconditioned dharmas differed but were always very short, comprising nirvana and sometimes space as well. But whether they were conditioned or unconditioned, dharmas were reality's ultimate simples, elements of reality which weren't further reducible to anything else, and were what they were because it was their nature to be whatever they are.
This kind of analysis could often be very dry and scholastic and some Buddhists rebelled against it. The Prajnaparamita literature began to preach universal emptiness, including 'dharma-sunyata', the emptiness of dharmas. The subsequent Madhyamaka philosophy tried to produce philosophical justifications for the Prajnaparamita teachings. Madhyamikas like the famous Nagarjuna were the ones who argued that anything with own-being (svabhava) must be eternal. So they basically attacked Abhidharma's whole dharma theory, insisting that nothing has an inherent nature that makes it whatever it is, arguing instead that everything becomes it is by causes, extending in an infinite regress into the past. Their motivation for arguing this way was was basically soteriological, based on the belief that anything solid in reality that could be emotionally grasped onto, would be grasped onto, and hence become a hinderance to enlightenment.
Whether fair or not, many Buddhists thought that Madhyamaka's doctrine of universal emptiness was nihilistic, and the philosophically absolute-idealist Cittamatra (mind-only) school appeared. As the name suggests, they insisted that there is something with svabhava, with it's own inherent nature, namely Mind. They continued to insist that reality was empty, except for them 'emptiness' has been reinterpreted to mean emptiness of subject-object duality. These philosophers argued instead that reality is merely a mental construct, and ultimately indistinguishable from the mind that constructed it. It's also important to note that this Mind that constructs reality isn't any of our individual minds, it's mind in general. So Cittamatra still hadn't abandoned the buddha's original insight that our individual selves are mental constructs and somehow illusory. For Cittamatra, our personal selves are still part of the universal illusion.
The similarity between Cittamatra and Advaita Vedanta is obvious. Some scholars think that how Shankara interpreted the Brahmasutras and in how he formed his own Hindu philosophy was strongly influenced by the Buddhist philosophical environment around him.