This is not a full answer but it was too long for a comment so I decided to write it as detailed answer (EDIT: I added what
I believe to be a full answer at the end resolving both points (1) and (2) in my previous answer).
I think it helps approaching this story from the other side. Instead of trying to find the correct formal group by staring at the formulas lets try to see if we can squeeze the formal group out from the topology.
Homology is represented by $H_*(-) = \pi_*(- \otimes H\mathbb{F}_p)$ from which we see (by virtue of the fact that $\pi_*$ is symmetric monoidal over a field) that the hopf algebra $\pi_* (H\mathbb{F}_p \otimes H\mathbb{F}_p) := \mathcal{A}_*$ must coact on $H_*(-)$. (i will write $\otimes$ for smash product of spectra throughout).
So homology determines a symmetric monoidal functor from spectra to graded $\mathcal{A}_*$ comodules. And in fact even before taking $\pi_*$ we still have (in some vague sense which might be difficult to make precise and I won't need) that $- \otimes H\mathbb{F}_p$ lands in the category of comodules over the "derived" hopf algebra $H\mathbb{F}_p \otimes H\mathbb{F}_p$.
Interpreting this algebro-geometrically we may think of $\pi_*(H\mathbb{F}_p \otimes H\mathbb{F}_p)$ as the ring of functions on some sort of "super" (which just means "in a graded sense") algebraic group $\mathbb{G} = Spec \mathcal{A}_*$ and of comodules over it as algebraic representations of $\mathbb{G}$. In this language homology determines a symmetric monoidal functor $\mathcal{Sp} \to Rep(\mathbb{G})$.
We may now precompose this functor with $\Sigma^{\infty}_+$ to get a symmetric monoidal functor (with respect to the product on spaces) $\mathcal{S} \to Rep(\mathbb{G})$ (where $\mathcal{S}$ stands for the category of spaces). Since every object in $\mathcal{S}$ is canonically a cocommutative coalgebra via the diagonal we immediately conclude that the functor actually lands in $CoAlg(Rep(\mathbb{G}))$. The algebro-geometric interpretation of coalgebras is as formal schemes by taking $Spf$ of the linear dual (there's a caveat here in that the filtration on the dual should come from the filtration on the homology coming from expressing a space as a filtered colimit of finite spaces). Moreover in this form the functor sends products to products since the categorical product in coalgebras is given by the tensor product on the underlying vector space.
Summarizing we now have homology as a functor $\mathcal{S} \to CoAlg(Rep(\mathbb{G}))$ which preserves products and filtered colimits. We can also compose with the fogetful $CoAlg(Rep(\mathbb{G})) \to CoAlg(grVect)$ which only remembers homology with the coalgebra structure. We may now use this to conjecture about the structure of $\mathcal{A}_*$ as follows.
An eilenberg maclane space $K(A,n)$ is a commutative group object in $\mathcal{S}$. Its homology is therefore a commutative group object in $CoAlg(Rep(\mathbb{G}))$. Interpreted algebro-geometrically the homology of $K(A,n)$ determines a commutative formal super group $\mathbb{G}_{A,n}$ together with a compatible action of the algebraic super group $\mathbb{G}$. In other words we get a map which we will hence forth call the comparison map:
$$\mathbb{G} \to Aut(\mathbb{G}_{A,n})$$
Where by $Aut$ I mean the full algebraic group of automorphisms of $\mathbb{G}_{A,n}$. We may now try to plug in values for $A$ and $n$ and check whether we get something valuable.
The most obvious first choice is $A = \mathbb{Z}/p\mathbb{Z}$ and $n=1$. It turns out that this brings us surprisingly close to the correct answer.
Let $V$ be an $\mathbb{F}_p$ vector space (equivalently a finite $p$-torsion abelian group) then there's the following computation:
$$H^{\bullet}(K(V,1),\mathbb{F}_p) =
\begin{cases}
Sym^{\bullet}(V^{\ast}[-1]) & \text{if $p = 2$} \\
Sym^{\bullet}(V^{\ast}[-2]) \otimes \bigwedge^{\bullet}(V^{\ast}[-1]) & \text{if $p \gt 2$}
\end{cases}$$
Algebro geometrically this can be interpreted as saying that $Spf H^{\bullet}(K(V,1),\mathbb{F}_p)$ is the formal super scheme corresponding to the super vector space $V[1] \oplus V[2]$ completed at the origin in the case where $p$ is odd and the completion of $V[1]$ when $p$ is even. It hence inherits the additive formal group structure from the addition on the corresponding vector space. Taking $V = \mathbb{Z}/p\mathbb{Z}$ we get something which one might call an additive formal super group.
Now comes the part i'm least sure about.
Its straightforward to compute the automorphism hopf algebra of $\mathbb{G}_{\mathbb{Z}/p\mathbb{Z},1}$. Then using the fact that we already know $\mathcal{A}_*$ and staring at it for a bit its easy to become convinced $\mathbb{G}$ is a sub algebraic-group of $Aut(\mathbb{G}_{\mathbb{Z}/p\mathbb{Z},1})$. There are therefore 2 questions left:
Find precisely the a subgroup of $Aut(\mathbb{G}_{\mathbb{Z}/p\mathbb{Z},1})$ isomorphic to $\mathbb{G}$. In other words what kind of extra structure on $\mathbb{G}_{\mathbb{Z}/p\mathbb{Z},1}$ do homology co-operation preserve?
Analyze the comparison map and determine whether it induces the inclusion from (1)
I think the for the even case we know that (1) is solved by requiring the automorphism be strict and that (2) is an isomorphism (I'm less sure about this point - so i'd love to be corrected about this if i'm wrong). For the case of $p$ odd I think the answer to (1) is that one should take automorphisms preserving the even coordinate to first order and the odd coordinate to first order modulo the even coordinate so it preserves the descending degree filtration on the tangent space. I'm less sure about this because I may have made a mistake in the calculation. Regarding (2) it's straightforward that the map must be injective since if a stable cohomology operation in mod p cohqomology is trivial on $B \mathbb{Z}/p\mathbb{Z}$ it must be the identity. And finally by computing the action on the cohomology using the axiomatic definition of the steenrod algebra we see that it's the same action.
EDIT:
Recently I discovered the story is actually not terribly complicated.
Let $CAlg_{\mathbb{F}_p}^{\mathbb{Z}/2}$ the category of graded commutative $\mathbb{F}_p$-algebras. Importing the definition from the ordinary case we we will define the additive formal "super" group as the formal group $\hat{\mathbb{G}}^{1|1}_{a}$ whose functor of points is:
$$R \mapsto (Nil(R),+) \in Ab$$
Where $Nil(R)$ is the nilpotent radical of $R$. This is of course the functor of points of $Spec H^{\ast}(B \mathbb{Z}/p)$ by the above discussion.
Similarly to the previous case we can define the functor of points of the automorphisms of $\hat{\mathbb{G}}^{1|1}_{a}$ as
$$R \mapsto Aut(Nil(R))$$
At this point in the $p=2$ all we needed to do was restrict to the subgroup of automorphisms which preserve the tangent space. Here however the tangent space is graded and the requirement that the derivative preserves the grading on the tangent space turns out not give the full dual steenrod algebra. The correct condition in the odd $p$ case will be that the automorphism must preserve the 2-step filtration on the tangent space (as observed by André Henriques in the comments). This means that the even part of the tangent space will be preserved and the odd will be preserved modulo the even. We now describe this in coordinates:
In coordinates a general automorphism $f$ (with no condition on the tangent space) of $(Nil(R),+)$ can be written in coordinates as
$$(f_+(x,\theta), f_-(x,\theta)) : Nil(R)^{\text{even}}_x \oplus Nil(R)^{\text{odd}}_{\theta} \to Nil(R)^{\text{even}} \oplus Nil(R)^{\text{odd}} $$
Where $f_+(x,\theta) = \Sigma_{n \ge 0} a_n x^{p^n} + \alpha \theta, f_-(x,\theta) = \Sigma_{n \ge 0} b_n x^{p^n} + \beta \theta$. We claimed that the correct "strictness" condition is $gr(df) = Id$ where $gr$ is the associated graded w.r.t. the $\mathbb{Z}/2$-filtration. In coordinates this condition translates to:
$$\frac{\partial f_+}{\partial x} = a_0 = 1, \frac{\partial f_+}{\partial \theta} = \alpha = 0, \frac{\partial f_-}{\partial \theta} = \beta = 1 $$
So the general form of a strict automorphism $f \in Aut^{\text{strict}}(\hat{\mathbb{G}}^{1|1}_{a})$ is $f_+(x, \theta) = x + \Sigma_{n \ge 1} a_n x^{p^n} , f_-(x,\theta) = \theta + \Sigma_{n \ge 0} b_n x^{p^n}$. Writing everything explicitly we see that the functor of points we defined is given by
$$R \mapsto \{ (f_+(x,\theta),f_-(x,\theta)) \in R[ [x] ][\theta]^{\times 2} : f_+(x, \theta) = x + \Sigma_{n \ge 1} a_n x^{p^n} ,$$
$$ f_-(x,\theta) = \theta + \Sigma_{n \ge 0} b_n x^{p^n} \}$$
This is a group for composition. Abstractly this functor can be defined as the functor of strict automorphisms of the $\mathbb{Z}/2$-filtered formal additive super group (where strict now means the same thing as before but we are now only enforcing it on the associated graded):
$$CAlg^{\mathbb{Z}/2} \to Fil_{\mathbb{Z}/2}(Ab), R \mapsto (Nil(R)^{\text{even}} \subset Nil(R),+)$$
The elements from the dual steenrod algebra define functionals on this group as follows
$$\xi_n : f \mapsto a_n , \tau_n: f \mapsto b_n $$
We can now check that the composition of power series induces a comultiplication on the dual steenrod algebra coinciding with the one we know. Therefore this map extends to an isomorphism $\mathcal{A}_{\ast} \to \mathcal{O}(Aut^{\text{strict}}(\hat{\mathbb{G}}^{1|1}_{a}))$. This resolves point (1) above. We can now resolve (2) as well by noting that the map above induces a comodule structure on $H^{\ast}(B \mathbb{Z}/p)$ which when dualized corresponds to the usual action of the steenrod algebra on $H^{\ast}(B \mathbb{Z}/p)$ via cohomology operations.
