Two possibilities:
The receptacle is incorrectly connected in series with the light (and if the light works from the switch, the receptacle is in parallel with that) instead of being connected directly to the power feed. This has probably never worked correctly since installation.
Or, much less likely, it's a multi-wire branch circuit (light on one hot leg, receptacle on the other) with a broken neutral, which puts the light and appliance in series across 240 volts. The fact the bulb didn't go pop suggests this is not the problem (tungsten would have blown immediately but LED might survive briefly). This could have worked correctly in the past until the neutral became disconnected.
We would need to see the wiring at the receptacle to make any progress. If it is incorrect, it suggests that whoever installed it didn't know what they were doing, therefore there might be other problems.