Let's start off with a closed transformer core, then open up an air-gap in it slowly, or add a secondary.
The flux in the core, the B field, swings over a range. For an low frequency iron core, it's typical to design a transformer so that range is about +/- 1.5T. It's the change in this B field that generates the back emf that opposes the input voltage. So with the input voltage a fixed amplitude (regardless of transformer loading) the B field has to swing over a fixed amplitude, again regardless of loading.
In order to magnetise the core to get that flux, there has to be a current flowing. In a transformer primary, with no load, the current needed to drive that flux round the core is called the magnetising current. That current creates an H field, ampere.Turns divided by the magnetic length of the core. With a closed iron core, having a relative permeability in the several thousands, not much H field is required to drive a sufficient B field.
When you let a current flow in the secondary, the secondary current opposes the primary current, which reduces the H field round the core, reducing the back emf. This allows more primary current to flow, until the H field again becomes strong enough to drive a B field that generates the same amplitude back emf.
If instead you open an air gap in the core, then the amount of B field you get for your H field drops dramatically, and again the B field amplitude falls, allowing more current to flow.
The effect of an air gap is quite dramatic. If we assume a fairly large transformer core, say 100mm x 100mm, the magnetic path will be in the order of 250mm long. If we assume transformer iron with a permeability of 2500, then that path length in iron is equivalent to 0.1mm in air. If we introduced a 0.1mm air gap, the magnetising current would double. If we introduced a 1mm air gap, it would increase by a factor of 11.
Once the air gap has reached the several milimetres scale, it stops being 'a gapped core', and starts being 'a coil of wire with a bit of iron in it', with the current dominated by the length of the air gap, not by the iron properties of the core.
A vibratory feeder will often be set up so there is as small as possible air gap between the core and the armature. This reduces the current required to drive it. The drive current remains very sensitive to the size of the gap.