Neither naturphilosophie nor vitalism (in some of its forms, anyway) maintained that there were things that cannot ever be explained by science. Indeed, they both were, or inspired, real scientific research programs. They both took the view that certain phenomena could not be understood through the sort of purely mechanistic theories popular amongst other scientists of the time, and that there might be other forces and modes of influence in nature, beyond those recognized by the science of their time. Naturphilosophie also emphasized the importance of holistic explanations, in terms of interactive process within systems, rather than an exclusive focus on the interactions of parts.
Naturphilosophie met with a lot of resistance and abuse from the more mechanistically inclined scientists of the time, and some of the ideas promoted in its name were indeed very vague, wooly and mystical. Nevertheless, in many respects, the naturphilosophers turned out to be right, and important aspects of their ideas became assimilated into the mainstream of science. Our understanding of electromagnetism and field theory owes a great deal to naturphilosophie. Ørsted and Faraday were strongly influenced by it: it was naturphilosophical **principles that led Ørsted to seek for, and eventually find, an interaction between electricity and magnets (a happy accident was also involved, but he would not have recognized its significance if he had not already been looking for such a phenomenon), and Faraday’s electromagnetic researches, and his concept of a field of force, also grew out of naturphilosophie.
Modern systems theory may not (so far as I know) actually have been directly inspired by naturphilosophie, but it is also broadly in accord with its principles, and inconsistent with the views of those who attacked it.
Van’t Hoff, the founder of stereochemisty (and, to a large extent, of structural organic chemistry as a whole) was viciously attacked by certain other chemists of his time on the grounds that his theories were influenced by naturphilosophie. Van’t Hoff was right, and his critics were wrong.
So far as I am aware, vitalism as such made less of a direct, positive contribution to our understanding of biology, but that does not mean that, in its heyday of influence, it was inherently anti-scientific. In the 19th and early 20th century, there were plenty of vitalist biologists doing real research and making real discoveries. They held that life could not be fully explained in terms of the principles and forces known to science at the time, but that did not prevent them either from furthering our understanding of how some aspects of life can be explained in terms of those known principles and forces, or from seeking for evidence and a scientific understanding of the “vital force” that they hypothesized to exist. The fact that it turned out not to exist, does mean that the search for it was unscientific.
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I don’t have an answer to the OP’s actual question. I think that there are indeed people who like to believe that some things can never be explained by science, but I do not know of a particular word or phrase that specifically characterizes that attitude. “Obscurantist” might cover it, but it is rather too broad.
In current philosophical jargon, those who maintain that consciousness (specifically) will forever remain beyond the reach of science are known as mysterians (which is actually derived from the name of the band who had a hit with “96 Tears”). However, most contemporary mysterians do not seem to want to believe that consciousness is scientifically inexplicable; rather, they hold that this is an unhappy conclusion to which they have been forced by the facts and logic of the situation. David Chalmers and Colin McGinn are probably the most prominent contemporary mysterians. They are not Cartesian dualists. I know Chalmers. In no way would it be fair to call him anti-scientific or obscurantist. He began his career in artificial intelligence research, and is knowledgeable and enthusiastic about modern neuroscience and cognitive science. I very much doubt that he is religiously motivated in his views. (I can detect no sign of religious motivation in McGinn’s writings either. Indeed, he makes it quite explicit that the he thinks materialism is true. He believes that consciousness had a material basis in the brain; he just does not think it is possible to prove this, or for the human mind to understand quite how consciousness arises from the brain.)
Not wanting some things to ever be scientifically explicable, is something completely different from the view that some things cannot be understood in terms of currently accepted scientific principles. Some of the people who have held the latter view, and have speculated about what other sorts of principles might be needed, have turned out to be wrong (which does not mean that they were necessarily silly or irrational), others came to be recognized as some of the greatest scientists of all time. Copernicus, Kepler, Galileo, Newton, Faraday, Planck, and Einstein all fall into this category.