Since the fuel pump in our car cycles on and off when you first turn the key on, would it cause any problems to hard wire the pump with a relay that turns it on solid once the ignition is on?

Clint

look up the fuel pump rewire on www.vfaq.com

In a word, no. People have been doing this for a long time.

If the ignition is on and the car is running, the pump never shuts off. When the key goes into accessory mode, the pump will kick on for a second to get the fuel pressure to normal ranges for cranking..