It is confusing to call this the “random poset”, as it is very different from what is usually called random posets in the literature: e.g., random posets have height 3 with high probability. Unambiguous standard descriptions of this structure are the countable universal homogeneous poset, the countable generic poset, or the countable existentially closed poset. (I will use the latter, as the axiomatization below rather directly corresponds to existential closedness.)

There is a perfectly general description. If $T$ is any universal theory in a finite relational language with the amalgamation property (and the joint embedding property, which however follows from AP in this case as long as you allow the empty structure), the Fraïssé limit of finite models of $T$ exists, and it is the unique countable existentially closed model of $T$. Its theory $T^*$ is $\omega$-categorical, and it is the theory of existentially closed models of $T$.

$T^*$ can be explicitly axiomatized by $T$ + all axioms of the form
$$\forall\vec x\:(\mathrm{Diag}_A(\vec x)\to\exists z\:\mathrm{Diag}_B(\vec x,z)),$$
where $A\models T$ is finite (possibly empty), $\mathrm{Diag}_A$ denotes (the conjunction of) the diagram of $A$, and $B\models T$ is an extension of $A$ of size $|A|+1$.

[Proof sketch: on the one hand, the Fraïssé limit satisfies the given axioms; this amounts to the defining property that any embedding of $A$ into the structure extends to an embedding of $B$. On the other hand, a straightforward zig-zag argument shows that the theory with the given axioms is $\omega$-categorical. Thus, it is the complete theory of the Fraïssé limit.]

This is essentially a special case of the (possibly infinitary) axiomatization of existentially closed models of a given $\forall_2$ theory in terms of *resultants*; see §8.5 in Hodges, *Model theory*, CUP, 1993.

In the case of $T$ being the theory of posets, the axiomatization of $T^*$ above simplifies to:

* The axioms of partial order.

* The axioms
$$\begin{multline}
\forall x_1,\dots,x_n,y_1,\dots,y_m,u_1,\dots,u_p\:\Bigl[\bigwedge_{i,j}(x_i<y_j)\land\bigwedge_{i,k}(u_k\nleq x_i)\land\bigwedge_{j,k}(y_j\nleq u_k)
\\\to\exists z\:\Bigl(\bigwedge_i(x_i<z)\land\bigwedge_j(z<y_j)\land\bigwedge_k(u_k\nleq z\land z\nleq u_k)\Bigr)\Bigr]\end{multline}$$
for all $n,m,p\in\omega$. That is, if $X,Y,U$ are finite sets such that $X<Y$ and no element of $U$ is below any element of $X$ or above any element of $Y$, then there exists $z$ such that $X<z<Y$ and $z$ is incomparable with all elements of $U$. Note that $n$, $m$, or $p$ may be $0$, thus the axioms ensure that the poset is upwards and downwards directed, but it has no minimal or maximal element.