Since the philosophy that deformation theory is controlled by DGLAs long predates the abstract characterisation of formal moduli problems, I'll break the answer in two. My answer will also hold in mixed characteristic, where things are complicated by the absence of a basepoints for the deformation functor (objects over $\mathbb{F}_p$ don't tend to have canonical deformations over $\mathbb{Z}_p$).
A) Deformation problems arising naturally tend to be governed by the SDCs of P-, Deformations of schemes and other bialgebraic structures,
Trans. AMS 2008, which is mostly contained in arXiv:math.AG/0311168. An SDC $E^{\bullet}$ is a diagram of formal schemes with an associative product
$$
E^i \times E^j \xrightarrow{*} E^{i+j},
$$
together with a unit $1 \in E^0$ and all the structure of a cosimplicial diagram except the outer coface maps $\partial^0, \partial^{n+1}$ (up to relabelling, this is the same structure as an augmented simplicial diagram); these satisfy various compatibility conditions. Cosimplicial formal groups give rise to SDCs, but do not exhaust them.
The Maurer-Cartan space of an SDC is then
$$
\{\omega \in E^1 ~:~ \partial^1\omega=\omega*\omega\},
$$
on which $E^0$ acts by conjugation.
In characteristic $0$, the homotopy categories of SDCs and of non-negatively graded DGLAs are equivalent, with equivalent Deligne groupids.
The constructions of P-, The homotopy theory of strong homotopy algebras and bialgebras, HHA 2010, arXiv:0908.0116 show how SDCs give rise to formal derived moduli stacks. Since SDCs have no negative terms, they can only give rise to derived $1$-stacks, but higher stacks can be obtained by incorporating additional simplicial structure.
B) In the proof P-, Unifying derived deformation theories, Advances 2010, arXiv:0705.0344
of the equivalence between DGLAs and formal derived stacks (later dubbed "formal moduli problems" by Lurie), most of the steps hold in positive or mixed characteristic.
This gives an equivalence between formal derived stacks and certain simplicial cosimplicial formal schemes up to tangent quasi-isomorphism. The relation with DGLAs is that in characteristic $0$, the simplicial and cosimplicial structures can be replaced with dg structures, and these become equivalent to Hinich's dg coalgebras, and hence to $L_{\infty}$-algebras via the bar construction.
