Teleology (the research of a phenomenon according to its "finality", its end-goal) is a subject that's often controversial (at least to my knowledge), and I'm not sure I've seen it in science (more specifically, natural science). Is there any teleological theories in (natural) science (more particularly, modern science)? If not, is there a fundamental rejection of teleological research?
Yes. There appears to be a consensus around teleology being present in the understanding of black holes. Checking for "teleology" in the arxiv might be illuminating and also remembering that until recently the least action principle used to bee seen as a kind of teleology.
A popular essay by Matt Visser asserts:
Teleology in physics has a very mixed score-card — the only known mainstream physics example is black hole event horizons...Black hole event horizons in general relativity are teleological; one has to wait till the trump of doom, and then back-track from the infinite future, to know whether or not anevent horizon is present right now.
(Matt Visser * From mindless mathematics to thinking meat?*)
The view is not uncommon as two other physicists point
Access to the complete space-time manifold is required to describe the black-hole. This feature necessitates that black-hole dynamics is specified by future or teleological boundary condition. Here we demonstrate that the statistical mechanical description of black-holes, the raison d'etre behind the existence of black-hole thermodynamics, requires teleological boundary condition.
(S. Bhattacharya, S. Shankaranarayanan, 2017, Black-hole event horizons-Teleology and Predictivity)
Vladislav Terekhovich (2015) discusses The Metaphysics of the Principle of Least Action and there are earlier treatments e.g. Stöltzner, M. (1994). Action principles and teleology, in Inside Versus Outside Berlin Heidelberg: Springer, p.33-62.
A telos in natural science could be taken as the evolution of an evermore refined being. However, with changes in environment the specific required characteristics may turn out different. So 'refined' in various circumstances may mean most efficient, skilled or resilient etc. Broadly, it could come down to saying the end of being is surviving, or being.
Applied to nature in general, the end of nature is 'to nature', that is to carry on doing what nature does.
So perhaps it becomes meaningless by way of being blindingly obvious.
I will throw in a quote from Derrida's The Ends of Man, page 13.
In the thinking and the language of Being, the end of man has been prescribed since always, and this prescription has never done anything but modulate the equivocality of the end, in the play of telos and death. In the reading of this play, one may take the following sequence in all its senses: the end of man is the thinking of Being, man is the end of the thinking of Being, the end of man is the end of the thinking of Being. Man, since always, is his proper end, that is, the end of his proper. Being, since always, is its proper end, that is, the end of its proper.