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To clarify, at this point the vehicle won't start right, but it seems like everything else electrical is ok?
The first thing I would check is the battery. It's not common that the battery dying would be enough to kill the engine, but subaru is picky about a solid voltage stream to keep the PCM (computer) running correctly. The battery acts as a voltage regulator. If the battery had a bad or weak cell it could be enough to turn on accessories on the vehicle (lights, radio, etc) but not enough to start the engine. You can test for that with a battery load tester. You will probably have to remove the battery from the car and take it to a parts store to have it load tested.
The next thing to check would be the alternator output. If the alternator has even slightly low output, in combination with a weak battery it could cause the engine to stall under a high voltage load (running the engine and the chair lift). The alternator output should be 14.9v or so with the engine running.
You might try jump starting the vehicle to see if it will start so you can then test alternator output. If that doesn't work then you will need to test voltages at the starter. It's hard to say what exactly to look for without knowing how the chair lift was wired in, but we can check the basics if we get to that point.
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