Most DLR area hotels would not offer a shuttle between the hotel and LAX (closer hotels to the airport do, but not to DL). Some hotels offer a shuttle between the hotel and DL (or you could use ART for that).
It's easy to book a shuttle or the DL Express from LAX to your hotel though. Taxi would be quite expensive (we were quoted $95/way). We much preferred the DL Express to a shuttle van though - took less time and more comfy.