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Let me pay no attention to size issues:

Denote the adjunction by $R$ right adjoint to $L$. Equip $\mathcal{E}$ with the canonical topology $J$ (so generated by jointly surjective epimorphisms), so that we have $Sh_J(\mathcal{E}) \simeq \mathcal{E}.$ Denote the induced sheafication functor $a:Set^{\mathcal{E}^{op}} \to \mathcal{E}.$ Now consider the Yoneda embedding $y:\mathcal{F} \hookrightarrow Set^{\mathcal{F}^{op}}.$ Since $\mathcal{F}$ is reflective in $\mathcal{E}$, it is cocomplete, so we can left-Kan extend the identify functor of $\mathcal{F}$ along Yoneda, to get a functor $a_\mathcal{F}:Set^{\mathcal{F}^{op}} \to \mathcal{F}$ which, by construction is left-adjoint to the Yoneda embedding.. EDIT: Since $\mathcal{E}$ is a topos, and therefore total, so is the reflective subcategory $\mathcal{F}.$ So we get to get a functor $a_\mathcal{F}:Set^{\mathcal{F}^{op}} \to \mathcal{F}$ which is left-adjoint to this Yoneda embedding. However, it is also canonically equivalent to $L \circ a \circ R_{!}$ since this composite is colimit preserving and along representables is the identity. So, it follows that $\mathcal{F}$ is a (Grothendieck) topos if and only if the composite $L \circ a \circ R_{!}$ is left-exact.

Note: By one of the comments I made below, $a \circ R_!$ is left-exact, so $\mathcal{F}$ is a topos if and only if $L$ preserves those finite limits of the form $\rho:\Delta_{aR_!(C)} \Rightarrow a \circ R_! \circ G,$ with $G:J G:D \to Set^{\mathcal{F}^{op}}$ a finite diagram.

2 added 278 characters in body

Let me pay no attention to size issues:

Denote the adjunction by $R$ right adjoint to $L$. Equip $\mathcal{E}$ with the canonical topology $J$ (so generated by jointly surjective epimorphisms), so that we have $Sh_J(\mathcal{E}) \simeq \mathcal{E}.$ Denote the induced sheafication functor $a:Set^{\mathcal{E}^{op}} \to \mathcal{E}.$ Now consider the Yoneda embedding $y:\mathcal{F} \hookrightarrow Set^{\mathcal{F}^{op}}.$ Since $\mathcal{F}$ is reflective in $\mathcal{E}$, it is cocomplete, so we can left-Kan extend the identify functor of $\mathcal{F}$ along Yoneda, to get a functor $a_\mathcal{F}:Set^{\mathcal{F}^{op}} \to \mathcal{F}$ which, by construction is left-adjoint to the Yoneda embedding. However, it is also canonically equivalent to $L \circ a \circ R_{!}$ since this composite is colimit preserving and along representables is the identity. So, it follows that $\mathcal{F}$ is a (Grothendieck) topos if and only if the composite $L \circ a \circ R_{!}$ is left-exact.

Note: By one of the comments I made below, $a \circ R_!$ is left-exact, so $\mathcal{F}$ is a topos if and only if $L$ preserves those finite limits of the form $\rho:\Delta_{aR_!(C)} \Rightarrow a \circ R_! \circ G,$ with $G:J \to Set^{\mathcal{F}^{op}}$ a finite diagram.

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Let me pay no attention to size issues:

Denote the adjunction by $R$ right adjoint to $L$. Equip $\mathcal{E}$ with the canonical topology $J$ (so generated by jointly surjective epimorphisms), so that we have $Sh_J(\mathcal{E}) \simeq \mathcal{E}.$ Denote the induced sheafication functor $a:Set^{\mathcal{E}^{op}} \to \mathcal{E}.$ Now consider the Yoneda embedding $y:\mathcal{F} \hookrightarrow Set^{\mathcal{F}^{op}}.$ Since $\mathcal{F}$ is reflective in $\mathcal{E}$, it is cocomplete, so we can left-Kan extend the identify functor of $\mathcal{F}$ along Yoneda, to get a functor $a_\mathcal{F}:Set^{\mathcal{F}^{op}} \to \mathcal{F}$ which, by construction is left-adjoint to the Yoneda embedding. However, it is also canonically equivalent to $L \circ a \circ R_{!}$ since this composite is colimit preserving and along representables is the identity. So, it follows that $\mathcal{F}$ is a (Grothendieck) topos if and only if the composite $L \circ a \circ R_{!}$ is left-exact.