'But what' is informal American English, and here has the meaning 'except that', 'without that' or 'but that'.
in American English
but that; but who; who or that … not
Who knows but what the sun may still shine
But what (Collins Dictionary)
A leaf fragment never blew under the door without that the wall panels
flipped open and the copper scrap rats flashed (rushed, luminously, perhaps, if they are copper) swiftly out.
A plainer, less poetic way of writing this might be
Every time a leaf fragment blew under the door, the wall panels flipped open
and the copper scrap rats rushed swiftly out.
Ray Bradbury (1920 - 2012) was a well known fantasy writer, famous for the fantastical and unreal themes in his writing. His writing style has been called 'poetic'. Wall panels that flip open releasing 'copper scrap rats' are definitely fantastical and unreal, and Bradbury's way of writing about them is decidedly poetic.
The 2026 house of 'the future' (as envisaged in 1950) is equipped with metal robot rats and mice which emerge from flaps in the walls to gather up scraps of waste and trash, including fragments of leaves:
The offending dust, hair, or paper, seized in miniature steel jaws,
was raced back to the burrows. There, down tubes which fed into the
cellar, it was dropped into the sighing vent of an incinerator which
sat like evil Baal in a dark corner.