I'm planning on claiming against my visa cards rental car insurance benefit. For an incident using a zip-car.
Will this claim affect my credit score or car insurance rates?
This would not affect your credit score at all. The claim itself, if made against your visa card rather than your car insurance, shouldn't directly affect your car insurance rates. However, if you were at-fault, you'll likely have to report that to your insurance agency and they may raise your rates as a result of that.