  We study critical Casimir forces (CCF) $f_{\mathrm C}$ for films of thickness
$L$ which in the three-dimensional bulk belong to the Ising universality class
and which are exposed to random surface fields (RSF) on both surfaces. We
consider the case that, in the absence of RSF, the surfaces of the film belong
to the surface universality class of the so-called ordinary transition.
  We carry out a finite-size scaling analysis and show that for weak disorder
CCF still exhibit scaling, acquiring a random field scaling variable $w$ which
is zero for pure systems. We confirm these analytic predictions by MC
simulations. Moreover, our MC data show that $f_{\mathrm C}$ varies as
$f_{\mathrm C}(w\to 0)-f_{\mathrm C}(w=0)\sim w^2$. Asymptotically, for large
$L$, $w$ scales as $w \sim L^{-0.26} \to 0$ indicating that this type of
disorder is an irrelevant perturbation of the ordinary surface universality
class. However, for thin films such that $w \simeq 1$, we find that the
presence of RSF with vanishing mean value increases significantly the strength
of CCF, as compared to systems without them, and shifts the extremum of the
scaling function of $f_{\mathrm C}$ towards lower temperatures. But $f_{\mathrm
C}$ remains attractive.
