Hi!
Do you have a local dish retailler/with their own installers? I frequently get returned receivers with useable pieces, and maybe for a small fee, they would keep their eyes open for a door for you. As far as pieces, you might want to look at auction sites and just find one that is listed "for pieces"...
But again, if it is just for aesthetic purposes, it's up to you. It really does have no point. If you have the pieces (I am going to assume that the little clips on the back of the door are now broken off), you could use a strong epoxy to glue them back on. Just be gentle with it from now on.
And if the door breaking is the worst that happened to it, you could just sit back and thank your lucky stars that it was just the door....LOL
Good Luck!
PS - if you are just concerned with keeping people's (kids" fingers out of it), try cutting down a thin piece of plexi-glass to fit the opening. If there is no card (internally wired and does not require a smart card), just use some light duty glue to keep things out of there....It can always be popped off if the need should arise at a later time, even if you do need a smart card. If you don't want to see through it, Krylon makes a great spray paint for plastics (brand name is Fusion)...Paint the back side of the plexiglass so it is still shiny on the "right" side when you install it....Hope any of this helps!