Perhaps you could go the other way? Prevent the peroxide from breaking down rather than forming more? Hydrogen peroxide is produced by every organism that uses oxygen in metabolism, and almost all of those organisms also have an enzyme called a peroxidase that breaks it down so it doesn't build up to dangerous levels in the organism. A virus that managed to deactivate the peroxidase enzymes would cause lethal buildups of peroxides.
For physical processes, to go from water to peroxide would require a strong oxidizer. Ozone is theoretically strong enough to accomplish that task, although I haven't actually seen a reference that it reacts that way. Perhaps a process that formed massive amounts of atmospheric ozone could generate substantial quantities of hydrogen peroxide from atmospheric water. If that process occurs, electrical discharges in moist atmospheres might also theoretically form hydrogen peroxide.
As for the implications on a world wide scale, there would be catastrophic effects long before a simple lack of water became an issue. Hydrogen peroxide is a strong oxidizing agent - anything that currently rusts or decomposes by physical oxidative processes would be sped up. Rubbers and plastics would rapidly decompose.