Can anions be reduced at the cathode in an electrochemical cell?
Theoretically, it is possible for anions to get reduced more. According to wikipedia's reduction potential table page, this reaction occurs at +0.08V vs SHE:

$\ce{S4O6^2− + 2 e− ⇌ 2 S2O3^2−}$

In your chloride example, though, you're trying to reduce $\ce{Cl^-}$ to (at least) $\ce{Cl^2-}$ and the potential for that reaction would be extremely negative. In most real-world scenarios, you would start reducing other things way before you hit that potential. Most likely you'd start reducing water at -0.8V.

$\ce{2H2O + 2 e− ⇌ H2(g) + 2 OH−}$

As an aside, be careful with your terminology. Reduction always happens at the cathode, that's what "cathode" means. Sometimes we get sloppy and don't switch the terms when we go from galvanic to electrolytic mode, but that doesn't change the chemistry. If electrons are flowing into an electrode, it's the cathode. I find the (+)/(-) labels on terminals more confusing than anything else.

Also, you said that


  for a galvanic cell, as electrons flow from the anode to the cathode, the cathode (+) becomes negatively charged due to the accumulation of electrons


This is backwards. The cathode does not become negatively charged because electrons have flowed into it. Rather, reduction occurs spontaneously at the cathode, leaving it electron-deficient. Electrons then flow from the anode to the cathode to compensate this charge imbalance and make both sides neutral again.