As a preamble, I consider it to be poor practice to use the term "supernatural" for the non-physical. "Natural" has a very clear meaning in epistemology -- it describes the items/events/forces which are subject to the tools of methodological naturalism -- IE empiricism and reasoning. Super-naturalism, from an epistemological perspective, are those things which one cannot apply reasoning and empiricism to. Use of "supernatural" for the ontologic category "spiritual" or for any non-physical thing, which are different but common uses, leads almost immediately into equivocation errors relative to epistemological meanings of natural and supernatural.
Note, there are many ontologically supernatural categories which are not epistemologically supernatural. All of Thomist philosophy and metaphysics, for instance, is purely rationalistic -- IE methodologically natural. And vitalism was a specialty within the science of biology at the start of the 20th century. And mathematics, logic, and ethics, do not appear to be physical fields, yet they are very much subject to methodological naturalism.
With that preamble, I will try to answer your question. There are two questions at issue. The first has to do with the ontology of our universe. Are there multiple fundamentally different types of things in it, or only one? And further, you are assuming that the existence of matter is beyond question, so the one is assumed to be material. What you are describing as "supernaturalism" would encompass a diverse set of ontologies, ranging from Platonic Idealism, Tegmark/Pythagorean math as source of everything, thru the Mind-centric idealist-leaning Perennial Philosophy, any concept of interactive spiritual dualism, Popper's strongly emergent consciousness, and probably Russellian Monism as well. Your "supernatural" just appears to be the postulate that there are non physical things or planes in our universe.
All of the above ontologic views assume that non-material things can be causal on the material -- IE that physics is not causally closed. This interaction, from outside physics, is generally described as "magic" by physicalists. "Magic" is the method by which the ontic supernatural interacts with the physical.
Physicalism is committed to the presumption that physics is causally closed, hence magic is impossible. Physics is not causally closed, and cannot be for multiple reasons.
- Physics is underdetermined, hence an outside influence can influence an outcome within that suite of options, entirely consistently with physics
- Physics, so long as it is an actual science, is by definition incomplete, as well as uncertain, so cannot exclude any phenomenon or outcomes (both ontic "supernatural" and "magic" are consistent with physics, so long as physics is not complete).
- No space within our universe, nor the universe itself, can ever be isolated from outside influence. Both entanglement and cosmology assume that no physical system can ever be completely isolated from outside influences.
- There are no absolute "laws" in science, only occasionally violated regularities, so causal closure, applied as a "law" is contrary to science.
Some principles of "magic", by which Platonic Forms influence their shadows in this world, or by which the Will to Power creates life events, or how Consciousness and Neurons both are reflections of the Russelian monod, are postulated or assumed in each of these differing ontologies. But most of the details are currently not filled it. Provided these "supernatural" ontologies are not epistemologically supernatural, than the application of methodological naturalism should allow the details to gradually get filled in as to how magic works in that ontology. Or else identify sufficient problems and/or contradictions that could lead to the abandonment of that ontology by most of its holders.
Note philosophy over the millenia has been subject to methodological naturalism, and subject areas and sciences have emerged from it to become their own science or academic specialty, as progress is made in characterizing how to evaluate that subject. "Magic" based "supernatural" ontologies are generally committed to this as a future path for their assumed ontology.
For an example of training a skill of "magic", IE applying methodological naturalism to refine a magic methodology, look a the methods developed by the CIA to perform remote viewing. Here is an example site that offers this training today: https://remoteviewingtraining.com/
For an example of a work that presumes the Perennial Philosophy as a starting point for doing science, and intrinsically accepts magic see Beyond Physicalism: Toward Reconciliation of Science and Spirituality https://www.amazon.com/gp/customer-reviews/RZY1A4EL2JOZ4?ref=pf_vv_at_pdctrvw_srp. This work treats "magic" as able to potentially operate through all 4 of the exceptions to causal closure of the physical I noted.
For one that takes a far more restrictive view of magic, and assumes that it will only act within the indeterminism of physics of the first exception, see Swinburne: Mind, Brain, and Free Will https://www.amazon.com/gp/customer-reviews/R18J8OJA7QPLKX?ref=pf_vv_at_pdctrvw_srp. Eccles also limits his magic to the indeterminate parts of physics: How the SELF Controls Its BRAIN https://www.amazon.com/How-SELF-Controls-Its-BRAIN/dp/3642492266/ref=sr_1_1?crid=XBJGKN0G1Q5X&keywords=eccles+how+the+brain&qid=1651905829&sprefix=eccles+how+the+brain%2Caps%2C214&sr=8-1
For an effort to understand how energy conservation could or could not apply to "magic" see this question and answer: 'The Zero Energy Hypothesis and its consequences for particle creation and dualist interactionism' https://physics.stackexchange.com/q/494408/181964
As to whether one must accept magic if one accepts "supernatural" ontology -- that one need not do so is the position of epiphenomenalism. Epiphenomenalism has a challenge to explain its own coherence, as asserting the truth of epiphenomenalism appears to be intrinsically a refutation of its own premise (assertions are physical, and presumably asserting the reality of consciousness as a non-physical item is a consequence of the reality of consciousness, hence is an examples of consciousness being causal on the physical). Despite the coherence difficulties, there are epiphenomenalists among current active philosophers. Both Chalmers and Jackson are "supernaturalists" relative to consciousness, and anti-magic epiphenomenalists relative to the effect of consciousness on the physical.