This is closely related to the discussion of pro-etale fundamental groups in my [paper](https://www.math.uni-bonn.de/people/scholze/proetale.pdf) 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](https://stacks.math.columbia.edu/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](https://www.math.uni-bonn.de/people/scholze/proetale.pdf). 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](https://arxiv.org/abs/2103.07305).