One of the criticisms of Aristotle's final cause category is that if a thing has a purpose, there must exist some entity that has intention to set up that cause. Generally speaking, skeptical thinkers reject any sort of positive, universal teleology on the grounds of Occam's Razor: until we can show that someone has designed the universe, we can't say that anything in it has a proper end or purpose. In a sort of reverse Teleological Argument, the suggestion is that without a purposer, there can be no purpose.
However, Aristotle certainly believed that teleological causes existed and sometimes were required to answer "why" questions. There's also no evidence that he believed in a creator or designer. The Stanford Encyclopedia of Philosophy suggests this passage as part of Aristotle's defense of final causes:
For these [viz. teeth and all other parts of natural beings] and all other natural things come about as they do either always or for the most part, whereas nothing which comes about due to chance or spontaneity comes about always or for the most part. … If, then, these are either the result of coincidence or for the sake of something, and they cannot be the result of coincidence or spontaneity, it follows that they must be for the sake of something. Moreover, even those making these sorts of claims [viz. that everything comes to be by necessity] will agree that such things are natural. Therefore, that for the sake of which is present among things which come to be and exist by nature. (Phys. 198b32–199a8)
(I encourage you to read the Encyclopedia's article on Aristotelian Causation to be able to follow the full argument.)
The upshot of the argument is that there is regularity in natural things (especially animals) because the regularity allows for them to survive or thrive. Having teeth of certain types in certain places allows animals to efficiently digest their food. Critically, Aristotle does not appeal to a creator or designer, but observes that final causes are embedded in nature itself.
Leaving aside the speculation of whether Aristotle would have rejected teleological explanations if he could have known of the Darwinian model, was his position valid? Can there exist positive, teleological explanations without the existence of a designer?