Presumably you are okay with a small drop, provided it is less than 0.5V.
Here is a circuit that will limit at 100uA and will drop less than 100mV before it limits.
simulate this circuit – Schematic created using CircuitLab
The MCP6001 is an inexpensive rail-to-rail input/output op amp that will operate from a 5V supply. The op-amp will saturate at ground until the load current reaches about 98uA nominally (with the values shown). The supply thus 'looks like' 5V with ~1K in series (the MOSFET contributes less than 10 ohms with Vgs =-5V), so it will drop between 0 and 100mV for load resistances of infinity down to 50K.
For lower load resistances the circuit regulates the output current to ~98uA.
The circuit draws about 200uA from the 5V supply, in addition to the load current of 0~100uA.