In other words you are asking about Pfaffian representations of a cubic hypersurface. To find a Pfaffian representation of a smooth cubic hypersurface is equivalent to constructing a vector bundle $E$ of rank 2 with $c_1 = 2$ generated by $6$ global sections. So, you are asking whether the moduli space of such bundles on a cubic surface is nonempty and connected.

As far as I know the moduli space is nonempty even for 3-dimensional cubics (and a fortiori for cubic surfaces). Also I know that in dimension 3 the moduli space is connected (in fact it is birational to the intermediate Jacobian of the cubic 3-fold).

In other words you are asking about Pfaffian representations of a cubic hypersurface. To find a Pfaffian representation of a smooth cubic hypersurface of dimension $d$ is equivalent to constructing a vector bundle $E$ of rank 2 with $c_1 = 2$ generated by $6$ global sections and with $H^\bullet(S,E(-k)) = 0$ for $1 \le k \le d$. So, you are asking whether the moduli space of such bundles on a cubic surface is nonempty and connected.

As far as I know the moduli space is nonempty even for 3-dimensional cubics (and a fortiori for cubic surfaces). Also I know that in dimension 3 the moduli space is connected (in fact it is birational to the intermediate Jacobian of the cubic 3-fold).