$\DeclareMathOperator{\Ext}{Ext} \newcommand{\G}{\hat{\mathbb{G}}} \DeclareMathOperator{\Maps}{Maps} \renewcommand{\phi}{\varphi}$ The analysis of the infinitesimal deformation space of the Honda formal groups $H_n$ uses three calculations which govern the existence of square-zero deformations. These can be phrased in terms of certain $\Ext$ groups, and here's how I think of them: we want to study the deformation of a formal group $G_0$ over a ground ring $R_0$ along a square-zero infinitesimal deformation $I \to R \to R/I = R_0$ to a formal group $G$ over $R$. Because $I$ is square-zero and the only formal group law truncated at degree $2$ is the additive formal group, we think of $G$ as sitting in an extension $I \otimes \G_a \to G \to G_0$ sort of "over" the original extension of $R$-modules. So, to study these extensions (their existence, their uniqueness, so on), we want to calculate some kind of $\Ext$ groups of the form $\Ext^*(G_0; \G_a)$.

This is exactly what Lubin and Tate do in their paper. Fix a pair of formal groups $F$ and $G$; then there's a simplicial object $BF = B_*(*, F, *)$ associated to $F$ via its group structure, and the cosimplicial object $\operatorname{Maps}(BF, G)$ should be a thing whose cohomology $H^*_{LT}$ computes $\Ext^{*-1}(F; G)$. Trying to work out exactly what this means, you'll find the following descriptions of the first few groups (written in coordinates, but this is inessential):

$$H^1_{LT}(F; G) = \{\phi: F \to G \mid \phi(x) - \phi(x + y) + \phi(y) = 0\} = \operatorname{Hom}(F, G),$$
$$H^2_{LT}(F; G) = \frac{\{\phi: F^2 \to G \mid \phi(x, y) - \phi(x, y + z) + \phi(x + y, z) - \phi(y, z) = 0\}}{\{\delta^1 \phi \mid \phi : F \to G, \delta^1 \phi(x, y) = \phi(x) - \phi(x + y) + \phi(x)\}},$$
$$H^3_{LT}(F; G) = \frac{\left\{\phi: F^3 \to G \middle| \begin{array}{c}\phi(x, y, z) - \phi(x, y, z + w) + \phi(x, y + z, w) - \\ \phi(x + y, z, w) + \phi(y, z, w) = 0\end{array}\right\}}{\{\delta^2 \phi \mid \phi : F^2 \to G, \delta^1 \phi(x, y) = \phi(x, y) - \phi(x, y + z) + \phi(x + y, z) - \phi(y, z)\}}.$$

General facts about infinitesimal deformation theory then tell you the utility of these groups: $H^1_{LT}$ tracks automorphisms of the square-zero deformation space, so it tells you whether you should expect the deformation space to be a scheme or some sort of stack; $H^3_{LT}$ tracks the obstruction to the existence of square-zero deformations; and $H^2_{LT}$ tracks the available square-zero extensions, in the sense that when $H^1_{LT} = 0$, its dimension will tell you the dimension of the tangent space of the deformation space.

So, Lubin and Tate go about computing these three things in the case that $G_0 = H_n$ is the height $n$ Honda formal group. They know that $\operatorname{Hom}(H_n, \G_a) = 0$, and so $H^1_{LT} = 0$. They also compute that $H^3_{LT} = 0$, so they know that the deformation space they're computing is actually a formal variety --- there are never any kind of conditions imposed on their generators $u_i$ to ensure the existence of a deformation. The only thing left is to determine the dimension of the formal variety, and they compute $\dim \Ext^1(H_n, \G_a) = n-1$.

You ask what happens when we replace $H_n$ with $\G_a$ and instead study $H^*_{LT}(\G_a; \G_a)$. The computation of these cohomology groups is worked out in the COCTALOS notes, and also in Theorem 4.3 and Section 8.4 of a project I worked on: [link]. Grinding through the homological algebra, you'll find that the group $H^n_{LT}$ consists of the polynomials of homogeneous degree $n$ in the dual Steenrod algebra, after assigning the degrees $|\xi_*| = 1$, $|\tau_*| = 1$, and $|P_*| = 2$.

All of these things are woefully $\infty$-dimensional. I don't actually know what that means for the existence of your object --- and so this isn't a proper answer --- but I do know that it isn't anywhere as simple as the case Lubin and Tate analyze.

P.S.: Since I'm updating this answer anyway, Section 2.2 of the Hopkins-Lurie *Ambidexterity* manuscript describes computing $H^*_{LT}(\Gamma; \G_a)$ for *any* connected $p$–divisible group $\Gamma$ and *any* value of $*$. This generalizes all but the last paragraph of this answer, since $\G_a$ is not $p$–divisible.