Calculating built-up density for each cell in vector grid in QGIS?

Question

I have two layers: a raster of India showing built-up areas (black pixels represent built areas) and a vector grid overlaying it. My goal is to calculate the percentage of built-up surface in each vector grid cell so as to better understand the urban density of certain regions (e.g some grid cells are 14% built-up, others 30% etc.) Other posts seem to suggest using zonal statistics, but this only provides calculations such as count, sum or min/max. Also having a hard time turning raster layer into vector as it is much too large (entire country of India).