You have to be careful: let $Y\subset V$ be a submanifold of $V$, you have a restriction map $$C^{\infty}(V)\rightarrow C^{\infty}(Y)$$ whose kernel is an ideal $p_Y$, and if $Y$ is a closed submanifold: $$C^{\infty}(Y)\cong C^{\infty}(V)/p_Y$$

Thus the question is rather what ideals of $C^{\infty}(V)$ are of the form $p_Y$ for $Y$ a submanifold of $V$.

1. if $Z$ is any subset of $V$ it makes sense to define the ideal $p_Z$ of functions of $C^{\infty}(V)$ that vanish on $Z$. And one can prove that a closed subset $Z$ of $V$ is a submanifold if and only if $p_Z$ is regular.

Now the question is what are the ideals of $C^{\infty}(V)$ of type $p_Z$ with $Z$ closed.

2. you consider $C^{\infty}(V)$ as a Fréchet space and notice that $p_Z$ is a closed ideal.

Hence the question is what are the closed ideals of $C^{\infty}(V)$ of type $p_Z$ with $Z$ closed?
Let me give you an answer when $V=\mathbb{R}^n$.

Let $I$ be a closed ideal of $C^{\infty}(\mathbb{R}^n)$ the quotient $C^{\infty}(\mathbb R^n)/I$ is called a differentiable algebra. $I$ is of the form $p_Z$ if this quotient algebra is reduced ($O$ is the unique element vanishing at any point of the real spectrum $Spec_r(C^{\infty}(\mathbb{R}^n)/I)$).

Reference: $C^{\infty}$-Differentiable spaces (LNM) Juan A. Navarro González, Juan B. Sancho de Salas

You have to be careful: let $Y\subset V$ be a submanifold of $V$, you have a restriction map $$C^{\infty}(V)\rightarrow C^{\infty}(Y)$$ whose kernel is an ideal $p_Y$, and if $Y$ is a closed submanifold: $$C^{\infty}(Y)\cong C^{\infty}(V)/p_Y$$

Thus the question is rather what ideals of $C^{\infty}(V)$ are of the form $p_Y$ for $Y$ a submanifold of $V$.

1. if $Z$ is any subset of $V$ it makes sense to define the ideal $p_Z$ of functions of $C^{\infty}(V)$ that vanish on $Z$. And one can prove that a closed subset $Z$ of $V$ is a submanifold if and only if $p_Z$ is regular.

Now the question is what are the ideals of $C^{\infty}(V)$ of type $p_Z$ with $Z$ closed.

2. you consider $C^{\infty}(V)$ as a Fréchet space and notice that $p_Z$ is a closed ideal.

Hence the question is what are the closed ideals of $C^{\infty}(V)$ of type $p_Z$ with $Z$ closed?
Let me give you an answer when $V=\mathbb{R}^n$.

Let $I$ be a closed ideal of $C^{\infty}(\mathbb{R}^n)$ the quotient $C^{\infty}(\mathbb R^n)/I$ is called a differentiable algebra. $I$ is of the form $p_Z$ if this quotient algebra is reduced ($O$ is the unique element vanishing at any point of the real spectrum $Spec_r(C^{\infty}(\mathbb{R}^n)/I)$).

Reference: $C^{\infty}$-Differentiable spaces (LNM) Juan A. Navarro González, Juan B. Sancho de Salas

Edit: @Nevermind, if you have a smooth surjective map $\pi:V\rightarrow Y$, then you will have a ring map $\pi^*:C^{\infty}(Y)\rightarrow C^{\infty}(V)$ and this map is injective.
Now I recommand you to look at Dominic Joyce's survey "Algebraic Geometry over $C^{\infty}$-rings" Corollary 3.4 He explains that the category of smooth manifolds embeds (fully, faithfully) as a subcategory of the category of finitely presented $C^{\infty}$-rings. Thus if you have a sub-$C^{\infty}$-ring $$R\rightarrow C^{\infty}(V)$$ this morphism of $C^{\infty}$-ring will be realizable by a smooth map $$V\rightarrow \mathfrak{R}$$ such that $C^{\infty}(\mathfrak{R})=R$ if and only if $R$ is an algebra of smooth functions and the way to recognize these algebras is exactly P. Michor's theorem in the note cited in your post. You notice that condition 1) of this theorem is satisfied for any subring of $C^{\infty}(V)$. Thus you are left with two criteria: "finitely generated" and "germ determined". Germ dertermined (related to condition 3) in Michor's theorem) is related to "fair $C^{\infty}$-rings" in D. Joyce's papers and related to reduced in the book on differentiable spaces cited above.

There is a very well written book whose title is "$C^{\infty}$-differentiable spaces" by Juan A. Navarro González, Juan B. Sancho de Salas (LNM springer). In particular chapter 2 to 5 may help you. For example one can characterize smooth submanifolds among closed subsets by looking at the associated ideal (not subrings):

"Proposition 2.9. Let $Y$ be a closed subset of a separated smooth manifold $V$ whose topology has a countable basis. $Y$ is a smooth submanifold of $V$ if and only if $p_Y$ is a regular ideal of $C^{\infty}(V)$."

As far as I know to characterize smooth manifolds among ideals we need:

1. to use the fréchet topology and work with closed ideals,
2. the quotient $C^{\infty}(V)/I$ can be viewed as a differentiable algebra and it should be "reduced" (see 3.4 of the reference cited above) ..... To be a little bit more precise if you consider $V=\mathbb{R}^n$, an ideal $I$ is of the form $p_Y$ if it is closed in the Fréchet topology of $C^{\infty}(\mathbb{R}^n)$ and if the space $Spec_r(C^{\infty}(V)/I)$ (which is a differentiable subspace of $Spec_r(C^{\infty}(\mathbb{R}^n))$) is reduced.

You should also have a look at Moerdijk-Reyes'book "Models for Smooth Infinitesimal Analysis" (section on ideals of $C^{\infty }(\mathbb{R}^n)$).

You have to be careful: let $Y\subset V$ be a submanifold of $V$, you have a restriction map $$C^{\infty}(V)\rightarrow C^{\infty}(Y)$$ whose kernel is an ideal $p_Y$, and if $Y$ is a closed submanifold: $$C^{\infty}(Y)\cong C^{\infty}(V)/p_Y$$

Thus the question is rather what ideals of $C^{\infty}(V)$ are of the form $p_Y$ for $Y$ a submanifold of $V$.

1. if $Z$ is any subset of $V$ it makes sense to define the ideal $p_Z$ of functions of $C^{\infty}(V)$ that vanish on $Z$. And one can prove that a closed subset $Z$ of $V$ is a submanifold if and only if $p_Z$ is regular.

Now the question is what are the ideals of $C^{\infty}(V)$ of type $p_Z$ with $Z$ closed.

2. you consider $C^{\infty}(V)$ as a Fréchet space and notice that $p_Z$ is a closed ideal.

Hence the question is what are the closed ideals of $C^{\infty}(V)$ of type $p_Z$ with $Z$ closed?
Let me give you an answer when $V=\mathbb{R}^n$.

Let $I$ be a closed ideal of $C^{\infty}(\mathbb{R}^n)$ the quotient $C^{\infty}(\mathbb R^n)/I$ is called a differentiable algebra. $I$ is of the form $p_Z$ if this quotient algebra is reduced ($O$ is the unique element vanishing at any point of the real spectrum $Spec_r(C^{\infty}(\mathbb{R}^n)/I)$).

Reference: $C^{\infty}$-Differentiable spaces (LNM) Juan A. Navarro González, Juan B. Sancho de Salas