So in Neukirch's Algebraic Number Theory book, page 343, he defines a formal o-module as:
A formal group $F$ over $O$ together with a ring homomorphism $O\to End_o(F), a\mapsto[a]_F(X)$, such that $[a]_F(X) = aX \mod{\deg 2}$
How is $[a]_F(X)$ defined?
Recall that an endomorphism of a formal group $F$ over $\mathcal{O}$ is simply a power series $f\in\mathcal{O}[\![X]\!]$ such that $f(F(X,Y)) = F(f(X),f(Y))$. Then a formal $\mathcal{O}$-module is the data of a formal group $F\in\mathcal{O}[\![X,Y]\!]$ together with a ring homomorphism $\mathcal{O}\to\operatorname{End}_{\mathcal{O}}(F)$ such that some condition is satisfied.
$[a]_F(X)$ is simply the notation of the image in $\operatorname{End}_{\mathcal{O}}(F)$ of $a\in\mathcal{O}$, and we require that it satisfy $[a]_F(X)\equiv aX\pmod{\deg 2}$. So $[a]_F(X)$ is a power series with coefficients in $\mathcal{O}$ such that there is no degree $0$ term and such that the coefficient of $X$ is $a\in\mathcal{O}$. There is not a specific definition given for $[a]_F(X)$, because we want to be able to consider potentially different formal $\mathcal{O}$-modules. That is, if $F$ and $G$ are two formal $\mathcal{O}$-modules, and $a\in\mathcal{O}$, it need not be true that $[a]_F(X) = [a]_G(X)$.