They never actually specify how the controls were selected, or what the odds ratios are they are computing.
If you think this is a valid reference, please write equations for the odds ratios and explain by what what criteria the controls were selected? Were they presenting as sick or randomly chosen from a data base?
However, this is a study of vaccine effectiveness, which is generally computed using odds ratios that assume the vaccines do not cause susceptibility to other conditions as a side effect. Thus they do not measure whether vaccines cause susceptibility to other conditions as a side effect.