N.B., in this answer I equate "non-physical causes exist" with "free will exists" since OP seems to have picked the latter as indication of the first. I have nothing much to say about non-physical causes that are not free-will/mind related... and as Stewart pointed out in his answer, what is a "cause" anyways...
TLDR: nobody knows, but there are strong opinions to be had.
But doesn’t recent science show that free will may be illusory
Absolutely not. There is a lot of philosophy out there which argues about free will and whether it is an illusion or not. Some of this philosophy takes modern scientific findings about certain measurements of brain activity into account to further its arguments. Free Will itself is more a concept than something physical - it's more a question about definitions and beliefs than something you could ever measure.
There is no science out there about this whatsoever, simply because science, while it has made good progress in brain research (i.e. how the visible or microscopic bits and pieces, the electricity, chemistry of the brain work, and how the whole mess may be structured into components or not...), it is awfully far away from producing any findings about how the mind works. Including questions about whether the mind, conscience, self and so on are a pure physical phenomena (i.e., illusions) or whether there is a different, non-physical aspect out there.
While it is easy - and getting steadily easier - to enjoy the "it's all physical" view these days, with many good arguments, there really is no falsifiable scientific evidence whatsoever; nobody has yet come up with a scientific, physical, experiment to prove the one or the other alternative.
It's eerily like the question whether gods exist: it is a question outside of science; there is simply no experiment out there that would prove or disprove the proposition. Like with god, unless you already believe in some super-physical aspect of our minds from the get go, it is very hard to find arguments supporting that assumption. And if you do believe in such, then it is very hard to find arguments dissuading from it.
So to your original question:
Do non physical causes exist?
Nobody knows, there are no experiments to prove or disprove it; there is no reason to believe they do. On the other hand, while there is no particular reason to believe that they do not exist, either; science is encroaching ever more and more and explaining ever more phenomena that are somewhere on the border between brain and mind, making it ever more likely that they do not exist (and if they do, then not in our minds at least; or to such a minuscule amount that it plays no role in everyday life).
On a tangent, the closest you can get to answer the question about free will and how the mind works is, as far as I'm concerned, not scientific, but simply self-reflection (in the form of meditation, e.g. Vipassana or similar purely observing, "insight" based meditations). You can, if you wish, view this as a kind of science, if you view yourself as the scientist; your own mind as the field of study; and yourself as the complete audience; and with unfortunately no way to extrapolate from your findings to something with relevance to other minds (i.e., no matter how long you meditate, you still have no way to solve e.g. the brain-in-a-vat question, or the living-in-a-simulation question; and whatever you personally tell me about your findings while meditating is nothing I can take for a fact about my mind, nor do I have any way to prove or disprove what you tell me about yourself).
On a tangent's tangent: while I personally find some of the science (i.e. that which you quoted about a limb moving before the "spark" has been measured in the brain) fascinating, I personally judge some of the purely philosophical arguments about the (non-)existence of Free Will much much more powerful. For example: one way - maybe the only way - to prove the existence of Free Will would be to "rewind the universe" to a point in the past, and check whether a person decided something different this time around. If a phenomenon requires such an absurd "proof", I strongly favor Occam's Razor.
Heck, and even if we were able to perform that experiment, and a person did indeed decide something differently, how would we ever know that it was not just random chance (i.e., physical quantum-based effects), which would be a very poor form of "Free Will" indeed.
Secondly, from personal experience, both while meditating and just while being mindful of what I do myself, I very much have the feeling that I would have decided any decision I had ever made exactly the same, if the universe were indeed rolled back. I personally would not be asking for science to disprove Free Will, but would challenge it to prove it (with my assumption that there will not be such a thing, ever); the same as I would not expect science to disprove God, but would rather challenge believers to prove it.