Icy moons would need oxygen to come down from their surfaces, where ultraviolet light and particle radiation spatters hydrogen out of the ice and off into space leaving oxidized molecules (not just oxygen) behind in the ice. This is possible, as on Europa the surface is young and it is believed to recycle surface ice down towards the internal ocean on megayear timescales (and Europa has an exceedingly thin oxygen 'atmosphere', one trillionth of a bar, from radiation-split water).

Figures I've seen (see "Energy, Chemical Disequilibrium, and Geological Constraints on Europa" by Hand et al) suggest that on Europa, a maximum of 5*10^9 moles of 'oxidants' may be delivered to the interior of Europa from the ice crust per year. Let's assume that's all oxygen - in that case, it's about a millionth of the photosynthetic oxygen flux of the Earth, and if we assume it is oxidizing hydrogen sulfide provides an energy flux of only about 45 megawatts to the interior.

That's less than a hundred thousandth the geothermal energy flux of the moon (and thus probably much smaller than the geochemical energy flux), but like the geothermal/geochemical energy flux it would not be even, there would be areas of downwelling ice with oxidizing agents slowly oozing out as it melted where this energy would be concentrated.