Did my baby get sick from breastfeeding?

No. Infections acquired from breast feeding are uncommon but not impossible. More likely, especially if by illness you mean a "cold" or vomiting or diarrhea illness, the illness was spread to the infant by hands or respiratory secretions during a cough or sneeze.
No. Breast feeding is the best thing that you can do to keep your baby healthy. If you catch a cold, the flu, or a bacterial infection your baby was probably exposed before you showed any symptoms. Your body begins to make antibodies to fight the infection, and the antibodies pass through your breast milk to the baby, to give him immediate 'infection protection' until he makes his own antibodies.