IgA-s in an immune system vaccined intramuscularly against Hepatisis A
Antibodies are produced after exposure to cognate antigen. 

Remember that B cells start off expressing membrane IgM and/or IgD. IgA requires class switching recombination at the B-cell receptor locus to the IgA isotype. Further differentiation of the B cells to antibody-secreting plasma cells results in soluble IgA production and immunologic memory to the antigen. Whether you'll get IgA, IgE or IgG antibodies against the vaccine will largely depend on the signals they get in the muscle, however (Figure 1). I think IgG is largely a result, since the tissue antiviral response is often interferon gamma (Figure 2). 

The importance of intramuscular vaccination is outlined here, just in case it wasn't apparent.



Figure 1. Activated helper T cells that can recognize the same antigen mediate the class switch by providing co-stimulatory signals and cytokines. Source



Figure 2. Class-switching continued. Source: Parham, The Immune System, 4th Ed.