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

No.

Sorry there isnt a better answer but as soon as it goes out of an unloaded chunk, it stops getting updated and will not move any longer. There are mods out there that can keep chunks active for a server so you might want to look into those, but not by default.

UPDATE: In the latest version of Minecraft there are devices called Hoppers. You can use these to keep chunks loaded and in theory can keep your track loaded so the cart would make it all the way. However Hoppers are NOT friendly to the processing speed of Minecraft. So this may have very adverse effects. However, if you want to try it out and see how it goes, you can hop over to this question to see how they are made: Can I use a hopper loop as a chunk loader?

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.