2000 Toyota Corolla will not start until hitting battery terminals

I have a 2000 Toyota Corolla which started to display a symptom during cold weather (-20 – -30 C) where you would insert the key, turn in partway, activate the internal electronics, and then turn it to start the car and it would just click once.

At first, just shaking the car by moving around in the seat seemed to bring the electronics back to life and then you could turn the key and the starter would go and fire the motor up.

Now, knocking on the battery or engine compartment seems to wake everything back up, and then you can start the car normally.

It seems like there is a loose wire or connection, but I’m not sure how the battery is wired up or what parts I might need to replace.