From my knowledge, gravity is infinite and extends throughout all of space. It diminishes as distance increases but is still present everywhere. So given enough time, no matter where something is in the universe, it would accelerate due to the gravitational force of something in the universe.
You must mean "the effect of gravity reaches to infinity" .
This is in classical mechanics, the law of gravity
We then have General Relativity,
General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 19161 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present.
Still classical physics up to here.
Also, from what I know, the plank length is the smallest possible length.
The Planck length is in the domain of quantum mechanics, ( h_bar is zero in classical mechanics) not classical mechanics. It is not the smallest possible length, since mathematics has no limits on variable sizes. It is a length small enough that the physics we know is undefined for lengths smaller than that.
And therefore wouldn't gravity have some sort of limit and not truly be infinite?
The force of gravity is mathematically infinite at r=0 according to the classical law of gravity. You mean that the effect of gravitational attraction should be limited at very large distances, though I do not understand your formula . It is at r=0 that quantum mechanics will take over and avoid the infinity, not at very large distances.
One will be able to talk of the relation , if any, of the Planck length to the law of gravity once one has a definitive quantized gravity theory, which is still a research topic.