I have two under dinette areas like that, mine already have doors.
What I did was go to Wal-mart and get two rightly sized under bed storage tubs. In my case the tubs are about 6" tall and extend most of the length and width of the space and just barely squeeze through the door opening. Way easier than a tray and adequate for this purpose.
I got something
like this tub and don't use the cover. If you look online there are loads of under bed storage tubs and boxes.