I believe the definition should be as follows:
Let $C$ be a Grothendieck site. Suppose $f:c \to d$ in $C$. Let $a$ be a cover $\left(a_i:d_i \to d\right)$ of $d$. Let $Sec(f)_a$ denote the set of maps $\sigma_i:d_i \to c$ is a map such that $f \circ \sigma_i=a_i$. Say that $f$ is a submersion if the set $\underset{a} \cup Sec(f)_a$, that is all maps $\sigma_i:d_i \to d$, ranging over all covers of $d$, is a cover of $c$.
It is easy to check that this gives the same definition you gave for manifolds.
EDIT: I made a mistake. What I actually should say is:
The collection $\sigma_i:d_i \to c$ defines a sieve $Sec$ on $c$, and we demand that there is some cover $h_j:c_j \to c$ such that each $h_j \in Sec$.