Short of keeping the vanity and cabinet doors open all the time(which I have done in the past), cut a hole somewhere in the side or door and cover it with a return air type vent. Then heat can get inside and to the pipes. Make a bigger hole in the vanity bottom and cover it with a vent also.
I'd bet that foam pipe insulation will be enough. Don't know about heat cord, but heat tape has a thermostat that keeps it from coming on till just above freezing and if your wall cavities are that cold then they need some small vents to allow heat inside anyway besides sealing off where the cold air is coming in from. You also need a receptacle for the heat tape/cord and wiring a gfci outlet under the vanity is too much trouble and you don't want to run an extension cord for this.
Lay a thermometer next to the pipe under the vanity and check your temperatures with the vanity door closed and then open to see what you are really dealing with.