Most of the answers are incorrect.
TimR's answer is correct.
I expect those assertions to meet with significant complaint.
:-(.

Here "the ingredients" means "ingredients required by the recipe", or, "ingredients which by the cooks revision of the recipe which will, in their estimate, achieve an acceptable result." The second meaning depends on the cook's skill and experience and makes discussion difficult. Assume the first definition for the following discussion. This can then be extended to the second definition if required.

To achieve "a pie" you need both all the essential ingredients AND enough of each to end up with something which is acceptably called a pie.
If it required 3 cups of flour and you had two cups of flour you could downsize it and adjust other quantities accordingly.

If it required a teaspoonful of salt and you had NO salt the result would be "a pie" but many would find it less acceptable than usual.

If it required baking powder and you had none then baking soda and an activator could be used by many cooks.

If it required walnuts on top you may be able to substitute hazelnuts.

But/and overall "the ingredients" certainly refers to a suitable set of components to build an acceptable finished product with. Having far too little of some and no effective substitute would qualify as "did not have (all) the ingredients".

Metaphor:

Never perfect, but useful.

"He will assemble an off road 4WD "Jeep" from the junk yard parts if it has the components. So, chassis, motor, gearbox, suspension, driveshaft, clutch, 3 road wheels, ... . NO. You can perhaps build an unusual 3 wheeler, but if you want a normal road vehicle you need 4 wheels.

...

Most of the answers are incorrect.
TimR's answer is correct.
I expect those assertions to meet with significant complaint.
:-(.

Summary:

• A pie is something you build to a recipe - which may be in your head but is often written down. You can improvise, but ultimately, there is a specification that you are building to. If you have SOME of all the ingredients but minimal flour or sugar or rising agent to the extent that you can't produce an acceptable product then "don't have the ingredients" is well understood short form English for the situation.
  NO competent cook with a somewhat below average or better grasp of English would query the statement. There are apparently a number of non-cooks here :-).

Here "the ingredients" means "ingredients required by the recipe", or, "ingredients which by the cooks revision of the recipe which will, in their estimate, achieve an acceptable result." The second meaning depends on the cook's skill and experience and makes discussion difficult. Assume the first definition for the following discussion. This can then be extended to the second definition if required.

To achieve "a pie" you need both all the essential ingredients AND enough of each to end up with something which is acceptably called a pie.
If it required 3 cups of flour and you had two cups of flour you could downsize it and adjust other quantities accordingly.

If it required a teaspoonful of salt and you had NO salt the result would be "a pie" but many would find it less acceptable than usual.

If it required baking powder and you had none then baking soda and an activator could be used by many cooks.

If it required walnuts on top you may be able to substitute hazelnuts.

But/and overall "the ingredients" certainly refers to a suitable set of components to build an acceptable finished product with. Having far too little of some and no effective substitute would qualify as "did not have (all) the ingredients".

Metaphor:

Never perfect, but useful.

"He will assemble an off road 4WD "Jeep" from the junk yard parts if it has the components. So, chassis, motor, gearbox, suspension, driveshaft, clutch, 3 road wheels, ... . NO. You can perhaps build an unusual 3 wheeler, but if you want a normal road vehicle you need 4 wheels.

...