1. Earth moving
Digging a hole into the ground doesn’t apply any net force on the Earth. But as you remove soil from the hole and pile it up into a heap, you are very slightly shifting the centre of mass of the planet. This will cause it to wobble a little as it rotates. The effect is very small, however.
2. Soil launching
If you fling each spadeful of soil into space, then you could apply a net force to the planet. You’d need to throw it hard enough to achieve escape velocity, but every billion tonnes you throw over your shoulder would change Earth’s orbital velocity by two nanometres per second.
3. Sea boiling
Alternatively, you could dig a hole at the bottom of the Pacific Ocean deep enough to drain all the water down into the mantle. If this boiled all the ocean into steam and jetted it into space, it might add an extra 1.5km/h to the Earth’s normal orbital speed of 108,000km/h.