As Ben Norris mentioned in his comment, ozonolysis is better described as a pericyclic reaction. Pericyclic reactions fall outside of the usual nucleophile-electrophile paradigm. This reaction is an example of a dipolar cycloaddition, because ozone is a 1,3-dipole. Chemists have observed that electron-rich alkenes react faster with ozone than electron-poor alkenes. This implies that the alkene is providing the HOMO to the reaction and ozone is providing the LUMO. In nucleophile-electrophile reactions, the nucleophile reacts through its HOMO, and the electrophile reacts through its LUMO. So if you want to make an analogy between a dipolar cycloaddition and a nucleophile-electrophile reaction, you could say that the alkene is the 'nucleophile' and ozone is the 'electrophile.'
It's not always obvious that the 1,3-dipole is going to provide the LUMO in a dipolar cycloaddition, but we can speculate as to why ozone does. Oxygen is a very electronegative atom, so its atomic orbitals are lower energy than carbons. Since ozone is made up of three oxygens (with low energy atomic orbitals orbitals), its pi-system will be low energy with respect to an alkene's pi-system. Furthermore, the pi-system will be even lower energy due to placing a cation on an electronegative atom. These factors contribute to the LUMO being low energy relative to an alkene.
To a first approximation, if there's a positively charged oxygen in a compound, it can be assumed that compound will be electrophilic.