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Let $F$ be a number field, $\mathcal{O}_F$ its ring of integers and $G$ an fppf $\mathcal{O}_F$-group scheme.

See the question Unramified Galois cohomology of number fields for unramified cohomology of algebraic groups over a field.

Is there an analogous definition of unramified cohomology classes in $H^1(\textrm{Spec}(\mathcal{O}_F)_{\textrm{fppf}}, G)$?

If we had a map \begin{align} H^1(\textrm{Spec}(\mathcal{O}_F)_{\textrm{fppf}}, G) \to H^1(\textrm{Spec}(\mathcal{O}_L)_{\textrm{fppf}}, G) \end{align} where $L = F_v^{\textrm{unr}}$ is the maximal unramified extension of $F_v$, we could say that a cohomology class is unramified at $v$ if it vanishes under this map. If this map exists, it shouldn't be too hard to define in terms of Čech cocycles, but I haven't had the time to think about this yet.

Here is some motivation. The sequence \begin{align} 1 \to \mu_n \to \mathbf{G}_m \to \mathbf{G}_m \to 1 \end{align} of fppf $\mathcal{O}_F$-group schemes gives rise to the descent sequence \begin{align} 1 \to \mathcal{O}_F^{\, \times}/\mathcal{O}_F^{\, \times \, n} \to H^1(\textrm{Spec}(\mathcal{O}_F)_{\textrm{fppf}}, \mu_n) \to \textrm{Cl}_F[n] \to 1. \end{align} If we can find some group of unramified $1$-cocycles containing the image of $\mathcal{O}_F^{\, \times}/\mathcal{O}_F^{\, \times \, n}$ and prove that it is finite, then we can prove a weak form of Dirichlet's unit theorem (the height part can easily be proved in analogy with the Mordell--Weil theorem).

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Nice question! This might not exactly answer it, but maybe covers the sort of thing you had in mind.

Regarding your motivation: the $n$-Selmer group of $F$, which I'll denote $\mathrm{Sel}_n(F)$, is defined as the subgroup of $F^{\times}/F^{\times n}$ consisting of classes represented by elements $x\in F^{\times}$ for which $(x)=I^n$ for some fractional ideal $I$. It naturally sits in a short exact sequence $$1 \to \mathcal{O}_F^\times/\mathcal{O}_F^{\times n} \to \mathrm{Sel}_n(F)\to \mathrm{Cl}_F[n]\to 1$$ as in your question, with the map $\mathrm{Sel}_n(F)\to \mathrm{Cl}_F[n]$ sending $x$ to $I$. The isomorphism $F^\times/F^{\times n}\cong H^1(F,\mu_n)$ arising from Kummer theory realises $\mathrm{Sel}_n(F)$ as the subgroup of $H^1(F,\mu_n)$ consisting of elements whose restriction at each nonarchimedean place $v$ of $F$ lands in the image of $\mathcal{O}_{F_v}^{\times}/\mathcal{O}_{F_v}^{\times n}\hookrightarrow H^1(F_v,\mu_n)$. For each place $v\nmid n$, this image just consists of the usual unramified cocycle classes, defined in terms of Galois cohomology. So you have a description of $\mathrm{Sel}_n(F)$ of the type you're after.

A note of caution though: the standard proof of finiteness of Selmer groups of elliptic curves that I know, e.g. as given in Silverman's book, essentially relates them to $\mathrm{Sel}_n(F)$ for suitable $n$ and $F$, and then uses Dirichlet's unit theorem, finiteness of the class group, and the exact sequence above to prove their finiteness.

To get closer to your actual question, the sequence of group schemes you consider shows that the image of $\mathcal{O}_{F_v}^{\times}/\mathcal{O}_{F_v}^{\times n}$ in $H^1(F_v,\mu_n)$ agrees with the image of $H^1(\mathrm{Spec}(\mathcal{O}_{F_v})_{\mathrm{fppf}},\mu_n)$ in $H^1(F_v,\mu_n)$. Proposition 2.2 of Česnavičius's paper (which contains other relevant discussions, results and references) then ensures that $\mathrm{Sel}_n(F)$ identifies with the injective image of $H^1(\mathrm{Spec}(\mathcal{O}_{F})_{\mathrm{fppf}},\mu_n)$ inside $H^1(F,\mu_n)$, and the two sequences are one and the same.

For general commutative finite flat $G$, the same proposition allows you to realise $H^1(\mathrm{Spec}(\mathcal{O}_{F})_{\mathrm{fppf}},G)$ as a 'Selmer group' inside $H^1(F,G)$. The subgroups $H^1(\mathrm{Spec}(\mathcal{O}_{F_v})_{\mathrm{fppf}},G)$ of `local conditions' identify with the subgroups of unramified cocycle classes for $v\nmid \#G$. One can see this by first comparing to etale cohomology, and then to unramified Galois cohomology.

Finally, it might be worth mentioning that Bloch--Kato Selmer groups (associated to e.g. $\mathbb{Z}_p(1)$, $T_p(E)$ for an elliptic curve $E$) give a natural framework that includes class groups, Selmer groups of elliptic curves, etc.

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