What is the universal deformation of $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$? Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of deformations is represented by a formal scheme isomorphic to $\mbox{Spf } \mathbb{W}(k)[[u_1,\ldots,u_{h-1}]]$.  Modulo lower terms, $u_i$ is the coefficient of $x^{p^i}$ in the $p$-series of the universal deformation.  (We take $p=u_0$.)
Does this carry over for the additive group?  Certainly there is an evident deformation to $\mbox{Spf } \mathbb{W}(k)[[u_1,u_2,\ldots]]$.  In the paper, the finite height assumption is present in various results that they cite from elsewhere, so without being intimately familiar with the whole theory it's kind of hard to tell if this assumption is essential.

This was beautifully and thoroughly answered at the MO link given in the above comment, so I'm marking this (community-wikily) as answered.
