The company EK uses in LAX is "Integrated Transportation Services" and the car they usually send (for J at least) is a Lincoln Town Car. It has a large trunk - which can take 4 suitcases comfortably, but if you are 4 J class passengers, it may be best to order 2 cars as suggested earlier.

But when they call to confirm the number of passengers and bags, you could ask for a minivan or a large SUV with more trunk space and they should be able to arrange it.