2000 Cadillac Deville Interior Knock

Hello propengr
Open the hood have someone start the car. Listen if the noise is from the engine compartement or under the dash or from the oil pan underneath.
Does the noise appears when the heater is on?
Check the engine oil, report level and condition.

Aise the car, start it while someone listen to the oil pan. I still suspect the oil pump. But it could be engine related.
A knocking sound is more mechanical than electrical,
there are electrical wires under the dash, and knocking sound. The oil pump is unerneath there.
It may be struggling when engine is cold.
If you hear the knocking sound while the air is turn off, so the problem is not coming from duckwork.
What do you mean by " increasing the vacum?"
when you give more gas?

Your hvac actuators are electrical, vacuum is from the days of old. Does the system function adequately, temperature changes(both sides), mode, and recirc functions seem normal? You only mention this knocking inside the vehicle when the car is started, which may be why an engine noise is suspect. Does it happen when the key is just turned to on?