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So the final answer, is 'no', but there is still something interesting to say:

Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top_{/\mathcal{T}}$ of topos $\mathcal{E} \rightarrow \mathcal{T}$ that are "étale over $\mathcal{T}$", i.e. of the form $\mathcal{T}_{/X}$. (This is the topos theoretic version of the representations of sheaves by their etale spaces)

As the category of Scheme indentifies in the way described in the question with a full subcategory $Sh(Et)$ this shows that morphisms of scheme $X \rightarrow Y$ corresponds exactly to morphisms between their étale topos, "over the étale topos of Spec $\mathbb{Z}$, i.e. geometric morphisms $f : Et_X \rightarrow Et_Y$ endowed with an isomorphism $f^* \mathcal{O}_Y \simeq \mathcal{O}_X$ as sheaves of rings, so they corresponds to a special kind of morphisms of ringed toposes.

One might wonder whether all morphisms of locally ringed toposes between étale toposes of scheme are of this form, but I'm affraind the answer is no. I'm quite far from algebraic geometry so I could be completely off, but I believe the following gives a counter example:

Consider $X = Spec \mathbb{Z}/p \mathbb{Z}$ and $\mathcal{T}$ its Gros étale topos.

Its structural sheaf is a sheaf of strict local $\mathbb{Z}/p\mathbb{Z}$ algebra, in fact the universal one.

I claim the Frobenius endomorphism of this sheaf provides a non-trivial non-invertible endormorphism of this sheaf: if it were trivial or invertible, then by universality, it would mean that for any sheaf of strictly local $\mathbb{Z}/p\mathbb{Z}$-algebra on any topos the Forbenius endomorphism would be trivial/invertible.

Hence the identity of the topos together with the Frobenius endomorphism provides a morphism of locally ringed topos that is not of the form above and hence do not corresponds to a morphism of scheme.

So the final answer, is 'no', but there is still something interesting to say:

Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top_{/\mathcal{T}}$ (where $Top$ is the 2-category of toposes) of toposes $\mathcal{E} \rightarrow \mathcal{T}$ that are "étale over $\mathcal{T}$", i.e. of the form $\mathcal{T}_{/X}$. (This is the topos theoretic version of the representations of sheaves by their etale spaces)

As the category of scheme indentifies in the way described in the question with a full subcategory $Sh(Et)$ this shows that morphisms of scheme $X \rightarrow Y$ corresponds exactly to morphisms between their (gros) étale topos, "over the étale topos of Spec $\mathbb{Z}$, i.e. geometric morphisms $f : Et_X \rightarrow Et_Y$ endowed with an isomorphism $f^* \mathcal{O}_Y \simeq \mathcal{O}_X$ between their structural sheaves (as described in the question) as sheaves of rings, so they corresponds to a special kind of morphisms of locally ringed toposes.

One might wonder whether all morphisms of locally ringed toposes between étale toposes of scheme are actually of this form, but unfortunately, the answer is no. I'm quite far from algebraic geometry so I could be completely off, but I believe the following gives a counter example:

Consider $X = Spec (\mathbb{Z}/p \mathbb{Z} )$ and $\mathcal{T}$ its gros étale topos.

Its structural sheaf is a sheaf of strict local $\mathbb{Z}/p\mathbb{Z}$ algebra, in fact the universal one.

I claim the Frobenius endomorphism of this sheaf (which is always a local morphism) provides a non-trivial non-invertible endormorphism of this sheaf: if it were trivial or invertible, then by universality, it would mean that for any sheaf of strictly local $\mathbb{Z}/p\mathbb{Z}$-algebra on any topos the Forbenius endomorphism would be trivial/invertible.

Hence the identity of the topos together with the Frobenius endomorphism provides a morphism of locally ringed topos that is not of the form above and hence do not corresponds to a morphism of scheme.

Edit: So I missed an important part of the question... I'm leaving a partial answer and I'm trying to find something better...

Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top_{/\mathcal{T}}$ of topos $\mathcal{E} \rightarrow \mathcal{T}$ that are "étale over $\mathcal{T}$", i.e. of the form $\mathcal{T}_{/X}$. (This is the topos theoretic version of the representations of sheaves by their etale spaces)

As the category of Scheme indentifies in the way described in the question with a full subcategory $Sh(Et)$ this shows that morphisms of scheme $X \rightarrow Y$ corresponds exactly to morphisms between their étale topos, "over the étale topos of Spec $\mathbb{Z}$, i.e. geometric morphisms $f : Et_X \rightarrow Et_Y$ endowed with an isomorphism $f^* \mathcal{O}_Y \simeq \mathcal{O}_X$ as sheaves of rings, so a special kind of morphisms of ringed toposes.

I don't think it is reasonable to expect that all morphisms of locally ringed topos between etale toposes of schemes would be of this form... But I'm not quite sure.

So the final answer, is 'no', but there is still something interesting to say:

Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top_{/\mathcal{T}}$ of topos $\mathcal{E} \rightarrow \mathcal{T}$ that are "étale over $\mathcal{T}$", i.e. of the form $\mathcal{T}_{/X}$. (This is the topos theoretic version of the representations of sheaves by their etale spaces)

As the category of Scheme indentifies in the way described in the question with a full subcategory $Sh(Et)$ this shows that morphisms of scheme $X \rightarrow Y$ corresponds exactly to morphisms between their étale topos, "over the étale topos of Spec $\mathbb{Z}$, i.e. geometric morphisms $f : Et_X \rightarrow Et_Y$ endowed with an isomorphism $f^* \mathcal{O}_Y \simeq \mathcal{O}_X$ as sheaves of rings, so they corresponds to a special kind of morphisms of ringed toposes.

One might wonder whether all morphisms of locally ringed toposes between étale toposes of scheme are of this form, but I'm affraind the answer is no. I'm quite far from algebraic geometry so I could be completely off, but I believe the following gives a counter example:

Consider $X = Spec \mathbb{Z}/p \mathbb{Z}$ and $\mathcal{T}$ its Gros étale topos.

Its structural sheaf is a sheaf of strict local $\mathbb{Z}/p\mathbb{Z}$ algebra, in fact the universal one.

I claim the Frobenius endomorphism of this sheaf provides a non-trivial non-invertible endormorphism of this sheaf: if it were trivial or invertible, then by universality, it would mean that for any sheaf of strictly local $\mathbb{Z}/p\mathbb{Z}$-algebra on any topos the Forbenius endomorphism would be trivial/invertible.

Hence the identity of the topos together with the Frobenius endomorphism provides a morphism of locally ringed topos that is not of the form above and hence do not corresponds to a morphism of scheme.

Yes :

As you point out, the category of strictly locally ringed topos is the category of $Top_{/Sh(Et)}$ where Top is the 2 category of topos and $Sh(Et)$ is the étale topos of Spec $\mathbb{Z}$.

Now given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top_{/\mathcal{T}}$ of topos $\mathcal{E} \rightarrow \mathcal{T}$ that are "étale over $\mathcal{T}$", i.e. of the form $\mathcal{T}_{/X}$. (This is the topos theoretic version of the representations of sheaves by their etale spaces)

So your question boils down to the (easy) fact that the functor from the category of scheme to the étale topos sending each scheme to its functor of point is fully faithful.

Edit: So I missed an important part of the question... I'm leaving a partial answer and I'm trying to find something better...

Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top_{/\mathcal{T}}$ of topos $\mathcal{E} \rightarrow \mathcal{T}$ that are "étale over $\mathcal{T}$", i.e. of the form $\mathcal{T}_{/X}$. (This is the topos theoretic version of the representations of sheaves by their etale spaces)

As the category of Scheme indentifies in the way described in the question with a full subcategory $Sh(Et)$ this shows that morphisms of scheme $X \rightarrow Y$ corresponds exactly to morphisms between their étale topos, "over the étale topos of Spec $\mathbb{Z}$, i.e. geometric morphisms $f : Et_X \rightarrow Et_Y$ endowed with an isomorphism $f^* \mathcal{O}_Y \simeq \mathcal{O}_X$ as sheaves of rings, so a special kind of morphisms of ringed toposes.

I don't think it is reasonable to expect that all morphisms of locally ringed topos between etale toposes of schemes would be of this form... But I'm not quite sure.