Fibrant in what model structure?
Simplicial abelian sheaves (and presheaves) are fibrant in the projective model structure because all simplicial abelian groups are fibrant.
Simplicial abelian sheaves are definitely not fibrant in the local projective model structure because sheaf cohomology groups can be nontrivial. For a specific example, consider the simplicial abelian sheaf Γ(Z[1]) on the site of smooth manifolds, given by applying the Dold-Kan functor Γ to the chain complex of sheaves Z[1] given by placing the constant sheaf Z in chain degree 1. If the simplicial abelian sheaf Γ(Z[1]) was fibrant in the local projective model structure, than the sheaf cohomology of a smooth manifold M in degree 1 with coefficients in Z could be computed by evaluating Γ(Z[1]) on M and then taking the homology group in degree 0. However, the homology group of Γ(Z[1])(M) in degree 0 is 0 for any M, a contradiction with the case M=S^1, for which H^1(S^1,Z)=Z.