If a 2 month old had an immune disorder would they be sicker?

Depends. There are an array of different immune issues and some are more severe than others. Those with severe immune disorders are often hospitalized with a major illness by 2-4 months.At that point their initial diagnosis is suspected and confirmed.
Sicker. Than what? Anyone with an immune disorder is going to be more prone to certain infections and conditions (depending on the nature of the immune disorder) than someone who does not have that disorder.