Does gravity turn itself off when it has nothing to work with?

The formula tells us that the force of gravity is equal to the mass of Object 1 multiplied by the mass of Object 2 multiplied by the Gravitational Constant and then divided by the distance between the objects, squared. It is implied, I think, that both objects have mass.

Correct so far as it goes. Physics is the scientific discipline which gathers observations of the behavior of matter and radiation and fits mathematical models to the data . The formula above is a very successful model of what happens when planets and satellites etc are involved.

There exists a more inclusive theory of gravitation though, expressed in fields, where all objects which have a mass have a gravitational field .

Does this mean that when Object 2 has no mass, or when Object 2 doesn't exist, there's nothing at all going on?

In this formalism, the gravitational field exists whether or not there exists a second mass to interact with it.

In order to measure the effect of a gravitational field one needs a second mass.

Any massive body M has a gravitational field g which describes its influence on other massive bodies. The gravitational field of M at a point r in space is found by determining the force F that M exerts on a small test mass m located at r, and then dividing by m:

$$\mathbf g(\mathbf r)= \frac{\mathbf F(\mathbf r)}{m} $$

This theory is very successful in describing our planetary system to great accuracy and inherent in its formulation is that the gravitational field is there, whether it is interacting or not.

Gravity is bound by the speed of light, it's not instant. That has a very real implication: if you have two objects a lightyear apart, it takes gravity a year to notice and react.

There's an implicit hidden mechanism involved in the question here: if gravity needs a trigger to start, then what starts the trigger force? It too would be bound by the speed of light, and that starter trigger would face another problem: it takes two years for the starter effect to travel from object 1 to object 2 and back if object1 would need to turn on its gravity for object2. So only after three years would object2 notice the gravity of object1. This would make gravity three times slower.