I found this video that shows the card transport. I'm not sure what this card tray is.
I don't know what a card tray could do other than get in the way. From what I can see - there IS no card "tray".
The card is simply held at the top edge by about 2mm. This should be quite adequate as the card does not weigh much and it will be
necessary to avoid fouling the various stations. The mag stripe station has a wheel to ensure proper contact and the embossing and topping are "sandwich" processes where the card only needs to be in approximately the right position and the mechanisms will clamp from both sides.
If it were me, I'd be looking much closer at the slot at the front of the card sled. Is there a clamping system or is it just a friction fit? I certainly would not be looking for a "tray" that I can't see being able to work.
This still does not explain the misfeed, as it seems the sled doesn't get close enough to pick up the card. Whether the sled movement is inadequate or the card feed mechanism isn't pushing the card back far enough, I don't know. Maybe it is a function of the thickness adjustment mentioned above.