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Let me add another answer addressing the first comment of the OP above, where he asks for possible hypothesis that ensure the orbit is locally closed. I hope I am not making a lot of mistakes. In summary, there is a positive answer to the question if one assumes that the scheme theoretic stabilizer $G_x$ of the section $x$ is flat.

$\textbf{Edit by afh:}$ Unfortunately this answer is not correct. I apologize, there is a small bug in one of the last steps in the argument below (the surjective morphism of flat schemes at the end does not need to be a closed immersion/isomorphism). In fact the statement of the proposition below is not true even if the base is a DVR and $G$ is etale and quasifinite. The proof below only shows that the set theoretic image is locally closed when the base is a DVR.

Let me add another answer addressing the first comment of the OP above, where he asks for possible hypothesis that ensure the orbit is locally closed. I hope I am not making a lot of mistakes. In summary, there is a positive answer to the question if one assumes that the scheme theoretic stabilizer $G_x$ of the section $x$ is flat.

This is what we have to show. Let $R$ be a DVR with field of fractions $K$. Suppose that we are given a morphism $Spec(R) \to X$ that factors set theoretically through the set theoretic image $\phi(O)$. The local valuative criterion stipulates that any section $Spec(K) \to O \times_{Spec(K)}X$ must extend uniquely to a section $Spec(R) \to O \times_{Spec(R)}X$. In order to check this, we are free to base change using the morphism $Spec(R) \to X \to S$ in order to assume that the base $S$ is the spectrum of a DVR.

This is what we have to show. Let $R$ be a DVR with field of fractions $K$. Suppose that we are given a morphism $Spec(R) \to X$ that factors set theoretically through the set theoretic image $\phi(O)$. The local valuative criterion stipulates that any section $Spec(K) \to O \times_{X} Spec(K)$ must extend uniquely to a section $Spec(R) \to O \times_{X} Spec(R)$. In order to check this, we are free to base change using the morphism $Spec(R) \to X \to S$ in order to assume that the base $S$ is the spectrum of a DVR.

We will use EGA IV (15.7.6). This Corollary in EGA says that if the valuative criterion for local properness (to be explained below) is satisfied for $ \phi: O \hookrightarrow X$, then the morphism $\phi$ factors as a composition $g \circ h$, where $h$ is an open immersion and $g$ is proper. In this case this would mean that $g$ is a proper monomorphism, and hence a closed immersion. In this (quasicompact) situation, this would in turn imply that $\phi: O \to X$ is a locally closed immersion. We are left to prove the valuative criteria mentioned above.

We will use EGA IV (15.7.6). This Corollary in EGA says that if the valuative criterion for local properness (to be explained below) is satisfied for $ \phi: O \hookrightarrow X$, then the morphism $\phi$ factors as a composition $h \circ g$, where $h$ is an open immersion and $g$ is proper. In this case this would mean that $g$ is a proper monomorphism, and hence a closed immersion. In this (quasicompact) situation, this would in turn imply that $\phi: O \to X$ is a locally closed immersion. We are left to prove the valuative criteria mentioned above.