In the introduction part of this renowned paper by S. Coleman and E. Weinberg, the authors write
The quartic self-coupling is required for renormalizability, to cancel the logarithmic divergence that arises in the amplitude for scalar Coulomb scattering.
The quartic term in scalar QED must be included in the Lagrangian (density) if one wishes to write the most general power-counting renormalizable Lagrangian (i.e., all non-negative mass-dimensional couplings) compatible with Lorentz invariance and gauge invariance.
The authors' statement although doesn't seem to contradict this but adds that if one neglects the quartic term, retaining only the quadratic mass term, the theory can become nonrenormalizable even with non-negative mass-dimensional term.
Do they mean massive scalar QED with vanishing quartic coupling is nonrenormalizable?