You didn't say how many miles you have on the car, but chances are if it stopped charging (the charge light is on), then the brushes are worn out. They are rather easy to replace. Since most mechanics replace alternators instead of repairing them, I highly doubt that they will have the brushes in stock, so you will have to special order them from your local Toyota dealer). Even at a dealer, I would be fairly surprised if the replacement brushes/holder would exceed $50.
You have to remove the alternator and remove the brush holder, then replace parts. The only other parts that normally go bad electrically are the voltage regulator and/or the rectifier - but when they go bad, there is usually other signs other than it's not charging. I've replaced lots of brushes, but never the other electrical parts in an alternator.
If you don't feel up to the challenge of "rebuilding" the alternator, you can usually find a local auto electrical shop that will do the rebuilding (they are the people that rebuild the ones that the auto parts stores are selling). If you take your old alternator in to them they can usually rebuild it and have it back to you in a couple of days for a lot less than buying a rebuilt alternator from a store.
If you rebuild it your self, or take the alternator someplace to get it rebuilt, you are going to have to remove it from the car. Otherwise, you are going to pay someone a fairly hefty labor charge to remove a couple of bolts (especially when they find out it's a Lotus $$$$)
I haven't checked (nor needed to remove my alternator) but since the alternator is mounted on the top side of the engine, chances are you remove the alternator from the top of the engine in the Elise. The diagrams below are from the Toyota Engine Manual, but they should be the same.
The alternator is only held on with a couple of bolts - you have to use a socket/wrench to let the tension off the serpentine belt, then remove the alternator. First diagram.
Once off, you start the disassembly. Second diagram is exploded view, third is the disassembly instructions to get you to the brushes.
Fourth diagram is how to check the brushes to see if they need replacing.
Fifth diagram is the installation instructions with torque values for the bolts.