Will a storage minecart keep going through chunks that aren't loaded?

There is no way to for a mechanism to pass through unloaded chunks or cause chunks to be loaded, but here's something you could try to make sure the minecarts eventually reach their destinations. I'm not certain whether it would work.

Arrange so that just before each chunk boundary, there is a downward-sloping powered rail. Then, as soon as the next chunk is loaded and the cart is able to move, it will move down the slope and into the loaded chunk.

If just being on a downslope isn't enough to prod the cart to move (unlikely), you could add a clock nearby to toggle the powered rail periodically.

you can set up nether portals in every chunk between them and have dispensers throw and item through it every 5 mins on a clock, ride it one and the chunks should never unload because the entity travels through the portal resetting the unload timer, you will need a lot of items though

With enough piston activity in a chunk it stays loaded. If you make a flying machine between 2 chunks (make sure part of it passes into the chunk next to it or an auto save will unload the chunks and stop the machine) all the chunks that it passes through will be loaded as long as its there. You can use this method to transport items if you put a minecart chest on it, however a server restart will still unload the chunks. This is solved by a sheep. Sheep still eat grass even if they are in a minecart. Put some grass with a named sheep in a minecart on top as well as a light so the grass grows back. When the sheep eats the grass, it will trigger an observer which can start the pistons back up. The only problem that I still need to solve is how to load the chunk with the sheep so it can eat the grass