Tienes [ extra para empezar en esta línea:-
->where([['DATE(shipping_table.sold_date)', '=',$date, ['shipping_table.shipping_status', '=' ,1]])
Tiene que ser (use DB::raw() para usar mysql functions ):-
$date = date("Y-m-d", strtotime($request->datepicker2));
$products = DB::table('shipping_table')
->select('products.product_name', 'products.price', DB::raw("Sum(shipping_products.quantity) as qtysold"), 'shipping_table.sold_date')
->join('shipping_products','shipping_table.shipping_id', '=', 'shipping_products.shipping_id')
->join('products','products.product_id', '=', 'shipping_products.product_id')
->where([[DB::raw("date(shipping_table.sold_date)"),$date], ['shipping_table.shipping_status', '=' ,1]])
->groupBy('products.product_name')
->paginate(8);
Nota:-
Con respecto al error que obtiene (Syntax error or access violation: 1055 ) verifique esta respuesta y haga lo que corresponda:-
aravel:error de sintaxis o infracción de acceso:error 1055
