Hare Krishnas say that Krishna stays beyond the material plane and interacts with the material world through his "expansions" if I understand correctly. Does Hindu scripture say anything like this?
No, that is not a Hindu concept.
God of Hinduism is present everywhere, even in the stones he is there.
See the following Rig Vedic Mantra:
adrau chidasmA antardurone (1), vishAm na vishvo amrtah swAdhih (2) ||
Even in the stone he is there for man, he is there in the middle of his house (1). He is as one universal in creatures; he is the Immortal, the perfect thinker. (2)
Rig Veda 1.70.4
God does not actively run the universe. It is primordial Nature, God’s shakti (power) which does everything.
Resorting to Prakrti, Nature, which is My own Power, I send forth again and again this multitude of beings that are without any freedom, owing to Nature's sway over them.
These activities do not in any way bind Me, because I remain detached like one unconcerned in their midst.
Under My direction and control, Nature brings out this mighty universe of living and non-living beings. Thus does the wheel of this world revolve.