Are there any 20/21th century philosophers that talk about a different approach to the study of nature other than science? Are there any that criticize science?
Of course! You know theologians exist right? They are the biggest critics of 'godless' science.
A big problem though is, how you define science. Several popes have said they accept science. The Dalai Lama says if any of the teachings of hus school of Buddhism are found in conflict with scientific findings, go with science. But these figures also hold many views which cannot be called scientific.
It might seem easy to define science, but I assure you it isn't. Falsifiability, double-blind controlled trials, repeatability, direct observations, and any number of 'hall marks' of science are actually very much discipline dependent. Occam's Razor or Baconian Induction exclude many things now accepted as science or as scientifically true. The idea science is unified in it's methodsvand homogenous is just totally wrong.
Popper developed his ideas on science specifically to stop Marxists saying historical materialism is scientific. It quickly gets political. And Dawkins and his ilk quickly resort to scientism, over truly scientific answers - which are after all very often, we don't and may never know; not a great crowd pleaser.
Critics of science from outside, tend not to have informed enough opinions to be widely listened to, and from inside tend to be reformers. The same holds generally for religions. James Lovelock or Lyall Watson argue passionately for expanded methods and paradigms, but from within, broadly, the community. I can only think of religious figures, and fringe ones, calling for a dismissal of scientific results.
Am thinking about how Foucault reinterpreted science in terms of power structures, and as influenced by and at least partly serving, biology, economics, and linguistics. So he certainly critiqued the idea that there is 'pure' science.
Thomas Kuhn's thinking about science suggested developments within it are at leastbto some extebt subjective and relative, which could be considered a rejection of ideas many scientists have about science.
Plenty of critics of aspects that may or may not be essential to science, like reductionism or positivism.
Shimon Malin’s “Nature Loves to Hide: Quantum Physics and the Nature of Reality, a Western Perspective” offers Whitehead as a 20th century philosopher that might be what you are looking for. Schrodinger and Heisenberg would be others.
His book attempts to explain the collapse of the wave function by linking fields of atemporal potentiality with the quantum event as an actualization. Science is limited by what Schrodinger considers “objectivation” which removes the subject from the observation. That can lead to what Whitehead calls the “fallacy of misplaced concreteness” where abstractions are confused with facts and are assumed to be valid outside their valid domains. In this way they criticize science.