In general, unless there is some intentionally isolated portions of a circuit, power supplies and circuits must all be referenced to a common point (called common or ground).
Put a small feedback capacitor of 1kpf from output terminal to inverting input to quench the noise .
As it is ,a common ground between the supplies must be connected to load ground requirement.
Great idea...
Unfortunately however, current will refuse to flow in the direction you need.
The 7805 (+5V regulator) IC only has current flowing OUT of the regulated terminal. Current cannot flow backwards through it.
Therefore two 7805 regulators cannot create a bipolar supply to power an op amp.
To provide a negative supply, you need to have current flowing into the regulated terminal.
This requires a P-device operating internally (whereas a positive supply uses an N-device internally).
This schematic will do about the equivalent job of a 7905 IC (-5V regulator):