Take the ice bucket out and see if the auger motor turns without it. You'll need to hold the door switch in when you press the dispenser lever. If the motor doesn't turn with the ice bucket out check for 120 Volts AC at the auger motor:
Auger motor 5304462594 / AP4299528 made by Frigidaire
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With a
DMM (Digital Multi-Meter). Access the auger motor and measure across the two wires on the motor. If your meter reads 120 Volts AC replace the motor. Less than 120 Volts AC check the ice dispenser micro switch for continuity with you meter.
Here's the front dispenser parts diagram for your model:
Parts for Frigidaire FRS6LF8FS1: Ice & Water Dispenser Parts in stock and ready to ship today. 365 days to return any part.
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Micro switch is #38A:
Dispenser switch 241689106 / AP6887337 made by Frigidaire
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There is a video in the Micro switch part link that shows you how to access it. Unplug your refrigerator first before you access it and ohm test it.
You can ohm the ice dispenser micro switch for continuity when you click it, Push-in should be continuity, Out should not be continuity.
Jake