Whenever an unbeliever starts inquiring regarding Buddhism or the Buddha's teaching, do Buddhists see this as providence leading the unbeliever to Buddha's faith or something else? Does Buddhism's notion of providence extend beyond karma?
Buddhists do not believe in a creator God nor in his or her divine providence. Buddhists believe in the law of cause and effect. When someone encounters the Buddhadharma it is because the causes of such a meeting are established and manifest.
There is no notion of divine providence and it is held that the law of karma is sufficient and necessary to explain how people encounter the Buddhadharma. Hope this answers your question.
Buddhism would say karma or volition (rather than an act of providence) brings people to Buddhism. This said, this kamma/volition is not always wholesome.
For example, a wholesome kamma bringing an individual to Buddhism is awareness of suffering and wanting to end this suffering. SN 12.23 says suffering is the supporting condition for faith.
An unwholesome kamma bringing an individual to Buddhism is believing the Buddha is like a god with unconditional love who loves you unconditionally, regardless of your unwholesome nature. In this situation, such an individual starts to believe their own unwholesome belief systems are what the Buddha taught.
I think there are elements of foresight -- "wisdom" features highly as a virtue; and if you're looking for parallels with the divine there are some elements of doctrine like Abhijñā (though I don't see that as central to the doctrine nor relevent to most practitioners).
In Pali, the language of the Buddhist scriptures, these four are known under the name of Brahma-vihara. This term may be rendered by: excellent, lofty or sublime states of mind; or alternatively, by: Brahma-like, god-like or divine abodes.
These four attitudes are said to be excellent or sublime because they are the right or ideal way of conduct towards living beings (sattesu samma patipatti). They provide, in fact, the answer to all situations arising from social contact. They are [etc.]
As for karma that's sometimes associated with "intention", and I suppose that "inquiring" is a consequence of various intentions -- to a large extent, the student's intentions. But it could also (since this is what you're asking) probably be traced back to the Buddha's intentions -- who, according to scripture, having found nibbana (nirvana), decides to teach: for the sake of those who will understand, see and hear.
And parents' and teachers'.
Can I tell you joke? I was told that, standing in a shop in Wales, someone was listening to two people talking Welsh -- and heard the word "efficient" used in a sentence. Apparently there's "no exact equivalent" for that word in Welsh! So they used the English word. Similarly, "unbeliever" sounded to me like a Christian term -- Chritianity, with its emphasis on "faith" -- or "infidel". Not that there is no Buddhist vocabulary (there is), but that word -- "unbeliever", rich in history -- struck me as having maybe "no exact equivalent".
I guess I also see it as a consequence of characteristics of the dhamma (the doctrine).