When answering a Physics.SE question, I made a claim that Newton realized that $F=ma$ worked in some frames, which are called "inertial frames." Nowadays, we know that there are non-inertial frames. One of the commenters suggested that started with d'Alembert. Further research shows he is certainly associated with the use of fictitious forces to explore motion in these non-inertial frames. But the harder question is the one I pose here: was Newton aware that in some frames, objects accelerate without forces being applied to them?
TL;DR Yes, but...
1) Inertial frames
To say that Newton had the modern conception of even inertial frames (based on the laws of motion), is an overstatement. Theoretically, he did not need them because he had absolute space, frames moving uniformly relative to it were inertial. In practice, he did have to spell out some of their properties to identify the fixed stars with such a frame.
But this is not to say that his successors did much better. Despite Euler's and d'Alembert's systematization of mechanical concepts at the conclusion of the vis viva controversy, according to SEP's The emergence of the concept of inertial frame, clarity on inertial frames did not come until the late 19th century. "Inertial systems" were independently introduced by Lange and James Thomson (Kelvin's older brother) around 1885, and the crucial condition that all accelerations must come from action-reaction pairs of forces attributable to bodies within the system, was added by Muirhead in 1887.
In hindsight, Mach in Die Mechanik in ihrer Entwickelung found implicit anticipation of these notions in Newton's treatment of orbital motions relative to the fixed stars, and in the Corollary V of Principia, which states that the center of gravity is unmoved by internal interactions, and remains at rest or in uniform motion in the absence of external forces. This is "as close as Newton could come to the notion of an inertial frame", according to SEP.
2) Non-inertial frames
What about non-inertial frames? Newton notes a special case of them in Corollary VI to the laws of motion in Principia, see SEP, “Quasi-inertial” frames: Newton’s Corollary VI:
"If bodies are moved in any way among themselves, and are urged by equal accelerative forces along parallel lines, they will all continue to move among themselves in the same way as if they were not acted on by those forces."
What Newton had in mind is discussed in his De Motu Corporum Liber Secundus (1687, the year Principia came out), where the Corollary is applied to the system of Jupiter's satellites and even, hypothetically, to the Solar system. However, in both cases the point was that the effects of such accelerative force are observationally immaterial:
"It may be imagined that the sun and planets are impelled by some other force equally and in the direction of parallel lines; but such a force (by Cor. VI of the Laws of Motion) would not change the situation of the planets among themselves, nor would produce any sensible effect."
3) Fictitious forces
Moreover, Newton did not make the connection between non-inertial frames and fictitious forces, let alone formulate anything like Muirhead's condition. For example, while his views of the centrifugal force evolved from the early years to Principia, there is no clear realization even there that it is only present in rotating frames (i.e. is fictitious). Bertoloni Meli in The Relativization of Centrifugal Force usefully summarizes Newton's handling of the centrifugal force as follows:
"In summary, before 1679 Newton - like Descartes, Borelli, and Leibniz - believed that orbital motion depended on the imbalance between gravity and centrifugal force; after 1684 he believed that centrifugal force was equal and opposite to gravity, from the third law of motion. In general, he explained curvilinear motion in terms of centripetal force and inertia alone, without centrifugal force; why in this case centrifugal force could be neglected, however, was not clear.
In certain passages, such as definition 5, Newton seems to associate centrifugal force with inertia. In other passages, such as the scholium to proposition 4, Book III, centrifugal force prevents an orbiting body from falling toward the center. In cases different from orbital motion, such as a planet rotating around its axis, he believed that centrifugal force was different from gravity. Hence Newton's theory of centrifugal force followed a case-by-case pattern, and the solution to one particular problem could not be easily generalized.
[...] For Huygens and Newton centrifugal force was the result of a curvilinear motion of a body; hence it was located in nature, in the object of investigation. According to a more recent formulation of classical mechanics, centrifugal force depends on the choice of how phenomena can be conveniently represented. Hence it is not located in nature, but is the result of a choice by the observer... Current views concerning the ideas on centrifugal force expressed by Newton in the Principia Mathematica are severely affected by the projection of modern methods and ideas that are found neither in the Principia nor in works contemporary with it."