The question leaves two details up to interpretation, and another that is precise may well be so by accident. This is important because the answer will be yes or no depending on how the details are made more precise. However, for most ways of making the details precise, the answer is yes.
- What do you mean by function? Apparently you are referring to functions from the reals (or possibly an interval - this doesn't really matter) to the reals. When mathematicians say "function" they usually mean functions (often implicitly restricted to functions from the reals to the reals) in the set-theoretical sense, which is the most general practical notion. But it seems you may have in mind a much more restricted definition, such as continuous functions, functions that can be described piecewise or even globally by power series, or even elementary functions.
- What do you mean by "can be found"? Some possible ways to make this precise: It is consistent with set theory that the antiderivative exists. It follows from standard axioms that the antiderivative exists. There is a constructive proof that the antiderivative exists. We can write down a power series for the antiderivative. The antiderivative is an elementary function.
- What do you mean by antiderivative? There is a precise, standard definition, but given your alternative explanation in terms of an integral it appears that that's not necessarily what you have in mind. There are many different definitions of integrals of varying generality (i.e. a non-continuous function may have an integral w.r.t. one notion but not another), and for all of them it makes some sense to refer to them as antiderivatives (in a wider sense). Fortunately, this detail turns out not to be relevant for the ultimate outcome, under reasonable assumptions on how the previous two points are made precise.
It is very well known that there exist functions from the reals to the reals which have a derivative $f$ that is not even continuous. (This is why many theorems use the words "continuously differentiable" in their assumptions, rather than the more general "differentiable".) Since a non-continuous function never has a derivative, any such function $f$ is an example of a function with an antiderivative but no derivative. But you may not consider this a real example because you may be interested only in continuous functions.
I believe some of these functions can be found in the strong sense that you can give precise definitions of them (and of their antiderivatives) - though not as power series, and certainly not as elementary functions. And the derivative cannot be found - in the strong sense that it definitely does not exist.
If you are only interested in continuous functions, then the answer is still yes. It is very well known, and easy to see, that there are continuous functions $f$ that are not differentiable. But every continuous function has an antiderivative. Again, this argument works with all definitions of "can be found".
If you are only interested in elementary functions, then the answer is no. It is well known that the derivative of every elementary function is an elementary function, but that there exist elementary functions whose derivatives are not elementary.