Here is a PyQGIS solution to save each band from a raster in a separate .TIF file and polygonize the results. To run it, all you need to do is copy paste the code into the QGIS Python Console and change the
import gdal,ogr, osr
in_path = 'C:\\Test\\Multiband Image.tif'
out_path = 'C:\\Test\\'
ds = gdal.Open(in_path)
raster_proj = osr.SpatialReference()
for i in range (1,ds.RasterCount+1): ##Begin at band 1 through to to the last one
#Save each band individually
srcband = ds.GetRasterBand(i)
out_ds = gdal.Translate(out_path + 'band' + str(i) + '.tiff', ds, format='GTiff', bandList=[i])
outShapefile = "C:\\Test\\polygonized"
driver = ogr.GetDriverByName("ESRI Shapefile")
outDatasource = driver.CreateDataSource(outShapefile+ str(i) + ".shp")
outLayer = outDatasource.CreateLayer("polygonized", srs=raster_proj)
#Add the DN field
newField = ogr.FieldDefn('DN', ogr.OFTInteger)
gdal.Polygonize(srcband, None, outLayer, 0, , callback=None )
sourceRaster = None
Depending on the number of bands and your data size, it could take a couple minutes. Moreover, I left out the merging of the vector layers, but you can merge them easily via the Merge algorithm in the Processing Toolbox.