Here's my solution which I derived from running through the Zonal Statistics plugin wizard and selectingviewing the generated Python command with Advanced -> ViewCopy as Python Code SnippetCommand. Then simply adding a loop to support the multiband requirement.
for i in range(1,num_raster_bands):
processing.run("native:zonalstatisticsfb", {
'INPUT':'my_shp_file',
'INPUT_RASTER':'my_raster_file',
'RASTER_BAND':i,
'COLUMN_PREFIX':'_',
'STATISTICS':[0,1,2],
'OUTPUT':f'output_filepath_{i}.csv'
})