Edit: I realize I answered a slightly different question, where $\mathcal F$ is just a sheaf, not a sheaf of abelian groups. The case of sheaves of abelian groups follows easily, for example by looking at the sheaves of isomorphisms with a given abelian group.
This is closely related to the discussion of pro-etale fundamental groups in my paper with Bhatt; that fundamental group classifies such $\mathcal F$.
So I claim that the answer is Yes. First, I claim that $\mathcal F$ is representable by a scheme $T\to S$ that is separated and etale (and satisfies the valuative criterion of properness). To see this, we use that separated etale maps descend along fpqc covers of the base, see Tag 0APK for descent of ind-quasi-affine morphisms (which separated etale maps are), and clearly the properties of being separated and of being etale descend. Thus, it suffices to show the claim fpqc locally on $S$, but then $T$ is just a disjoint union of copies of $S$.
The image of $T\to S$ is necessarily open and closed; away from it, $\mathcal F$ is empty, so we can assume $T\to S$ is surjective. Then $\mathcal F$ is trivial after pullback along the etale cover $T\to S$.
It turns out that for general $S$, it is slightly tricky to characterize the class of $\mathcal F$ that are fpqc locally trivial. By the above, all of them are representable by schemes separated and etale over $S$, satisfying the valuative criterion of properness. If $S$ has locally a finite number of irreducible components, the converse is true, see Lemma 7.3.9 and Remark 7.3.11 here. In general, the following may however happen: There is some $\mathcal F$ that is not fpqc locally trivial, but for which $\mathcal F\sqcup \mathbb Z$ is fpqc locally trivial, see Example 7.3.12 in loc.cit.
Another, more relevant, word of warning: It is not true that a general dualizable group scheme is trivial after a finite etale cover. This is only true for normal schemes; it fails for example for $\mathbb P^1$. Interestingly, it's equivalent to admitting faithful representations, see here.