You can certainly drop a minecart through an End portal, but the core problem I suspect you're going to run into is that any time an entity enters the End, it clears out the 5x3x5 area above the spawn platform, so that's all your rails gone. So while you can't have it land directly on a track, you do have some workarounds:

• Add a passive iron and bamboo farm near your other farms and automatically manufacture chest minecarts on-site. Tedious to set up and it doesn't actually recycle the minecarts, but of the available options it has the fewest points of failure.

• Build some complicated mechanism to manually assemble a rail system every time a minecart comes through the portal and then disassemble it after it leaves, and just hope it doesn't get erased itself. Don't do that one because it's dumb and will break but it's fun to think about.

• Use a water stream to push the minecart off the spawn platform and onto a rail; the water will get erased every time something passes through the portal, but if you put the water sources beside the spawn platform it'll automatically refill itself. Probably the most reliable way to cycle minecarts, but you will have to turn your End spawn into a wading pool.

• Entities maintain momentum when passing through portals, so if you shoot the cart through the portal fast enough you may be able to just wing it directly onto a rail just off the spawn platform. It'd take a bit of aim to get the speed and trajectory lined up right, and even then minecart physics are inconsistent on a good day, but you won't have to get your shoes wet.