This is not a direct answer to the question for a general group scheme $G \to S$ and I am not an expert in this area. However, I would like to point out that the resolution property of stacks is a natural condition that appears in this context of Hilbert's 14th problem by work of R. W. Thomason:

*Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes
Advances in Mathematics 65, 16-34 (1987)*

Once and for all let $ \pi \colon G \to S$ be an affine, flat, finite type group scheme over a noetherian and separated base scheme $S$.

Recall, that a noetherian algebraic stack has the *resolution property* if every coherent sheaf is a quotient of a vector bundle (a locally free sheaf, which will be always assumed to be of finite and constant rank).
Therefore, the classifying stack $B_S G$ has the resolution property if and only if every coherent $G$-comodule on $S$ is the equivariant quotient of some locally free $G$-comodule. The latter is the definition of the *$G$-equivariant resolution property* of $S$.

What we need is his Theorem 3.1:
$G \to S$ can be embedded as a closed subgroup scheme of $GL(V)$ for some vector bundle $V$ on $S$ if $B_S G$ has the resolution property. If $S$ is affine, $V$ can be taken to be free.

Thomason does not say that the converse to Theorem 3.1. also holds. I guess that this is true if $S$ is affine, but as I am always getting confused while working with comodules, I cannot give a rigorous proof at the moment.

Nevertheless, it is worth to ask when $B_S G$ has the resolution property. Thomason proved this in the following cases:

- $S$ regular and dim $S \leq 1$,
- $S$ regular; dim $S \leq 2$; $\pi_* O_G$ is a locally projective $O_S$-module, e.g, if $\pi \colon G \to S$ is smooth and with connected fibres.
- $S$ regular or
**affine** or has an ample family of line bundles; $G$ a reductive group scheme which is either split reductive, or semisimple, or with isotrivial radical and coradical, or over a normal base $S$.

In particular, if $S$ is the the spectrum of the ring of dual numbers, then this provides an affirmative answer to the posted question if $G \to S$ satisfies the conditions in (3).

Even for $G \to S$ arbitrary with reduction $G_0 \to S_0$, we know that the reduction $X_0=B_{S_0}G_0$ of $X= B_S G$ has the resolution property by (1).
So we may reformulate the original question as follows:

(Q2) Is the resolution property preserved under the first order deformation $X_0 \to X$?

Lifting of various locally free resolutions from $X_0$ to $X$ is probably not the best approach. However, it suffices to lift a *single* locally free sheaf.
Let us see, why this is true.
A noetherian algebraic stack with affine diagonal has the resolution property if and only if there exists a vector bundle $V$ whose associated frame bundle has quasi-affine total space.
The normal case was proven by Totaro in

*The resolution property for schemes and stacks.
J. Reine Angew. Math. 577 (2004), 1--22.
14A20 (14C35)*

and in my thesis, I am currently working on, I show that this really holds for non-reduced stacks too.
Therefore if we can lift $V_0$ from $X_0$ to a vector bundle $V$ on $X$, then $V$ has still quasi-affine frame bundle as its reduction is quasi-affine.
The obstruction for this lies in $H^2(X_0, I \otimes V_0^\vee \otimes V_0)$ where $I$ is the coherent ideal of order two defining the deformation $X_0 \to X$. Probably, the ideal can be removed here with some tricks.

In our case this cohomology boils down to the second group cohomology of the $G_0$-representation $I \otimes V_0^\vee \otimes V_0$. In particular, if $G_0 \to S_0$ is linearly reductive, the obstruction is zero.

Therefore we have proven:

If $G \to S$ is a group scheme over an artinian base with linearly reductive special fibre, then $G \to S$ can be embedded into some $GL_{n,S}$ as a closed subgroup scheme.

Clearly this still leaves out interesting cases and probably this can be proven more directly avoiding stack theory.