Positrons can come from a number of sources.
However, in PET they are all produced by nuclear decay. Basically unstable nuclei are produced in a cyclotron by bombarding target material with protons. A typical reaction is to have a bombarding proton enter the nucleus of the target material and kick a neutron out in the process. Changing the number of protons in the nucleus changes the atomic species and in this case the atom is changed from oxygen to fluorine. The new nucleus created in this manner is unstable and eventually decays into a more stable form.