Disclaimer: At first, I want to highlight that I am not familiar with AP Calculus and the corresponding syllabus, so I will cover solely the mathematical dimension of this question.
So, teaching calculus without limits, in general, means the you have at hand some other means of talking about approximations etc. The only way that comes to my mind to do so is by using infinitesimals - i.e. positive quantities $\varepsilon$ such that $\varepsilon<x$ for any real $x>0$.
So, using this and standard parts of hyperreal numbers, one may define the derivative of a function as follows:
The derivative of a function $f$ at some point $a\in D_f$ is defined as the standard part of the following hyperreal:
whenever this standard part exists/has sense.
However, I find it mostly improbable that this was the one preferred in an introductory calculus class. However, one may "define" a notion as e.g. the derivative, procedurally by describing the way one may find a functions derivative - in terms of algebraic manipulations.
For instance, one way to do so is to take some $h\neq0$ and then calculate the rate of change fraction:
Then, once the above fraction has been simplified enough, by setting $h=0$ - just to avoid to mention any limits - you get the so wanted derivative.
Even if the above approach has several - and from my viewpoint, enormous - faults in terms of sacrificing mathematical rigour in favour of a more procedural understanding, it still seems to me the only "reasonable" way in which one may avoid mentioning limits in a calculus class.