No, it won't work when Vdd is 0V, the MOSFETs need a bias voltage to keep them open.
As Olin says it would help to know exactly what you are trying to do to determine the best solution, but for an electronic normally closed switch, here is a simple idea:
Most MOSFETs are enhancement mode, which means the MOSFET is off with 0V gate-source bias (Vgs) and turns on with a positive Vgs bias (for a N-ch, opposite for a P-ch)
What you need in this situation is a depletion mode device, which means with 0Vgs, the FET is on, and turns off with a negative Vgs (assuming N-ch again)
A typical JFET is a depletion mode device, and you can also get hold of depletion mode MOSFETs such as the BSS139.
So using something like the above, here is a simple circuit (that could be elaborated on if necessary):
Ignore the resistor R3, this is just to simulate a switch by setting it from low to very high impedance - the SWITCH node would be connected to your bias voltage needed to turn the FET off (so it's connected to -10V in this example)
Simulation:
Above we can see the SIG_OUT when the JFET gate is left floating (Red trace) and then when biased with -10V (Blue trace)
The signal in is 200mV pk-pk with a DC offset of 0V, so this can be used for dual polarity signals. Depending on the JFET used, the gate does not have to be biased so low, the smaller the Vgs required to turn it fully off the better.
Note that the ON resistance of this switch will be quite high, so you cannot load it too much - if you need to drive something then you will need a buffer in between.
If you don't want to use a negative rail, the same concept can be applied to a P-ch JFET:
I haven't included the simulation as it's exactly the same as above. The bias voltage used was floating (e.g. if using a switch on the gate it's open) and +10V to turn off (so switch would be wired to +10V)
The FET part numbers shown can obviously be changed if desired, I'm sure there are better parts out there - they were just picked from the small selection LTSpice has.
You can add a parallel path with a transistor for current to flow from your 5V rail through the LED to ground. Effectively you're building an OR-gate, so the LED will be on if the button is pushed OR your micro-controller decides it wants the LED on.
Here's an example schematic using a N-channel MOSFET. The MOSFET is on when the IO pin is driven high, and it's off when the IO pin is driven low.
Best Answer
If the button is normally open (no connection when not pressed) you can solder two wires at the bottom to the switch contacts and feed those to the circuit which needs to control the 'button'. The button itself can remain in place. There is no need to de-solder it.
You can use a relay which is the simplest and safest method as that will work independent of how your fan controller works. If the button needs to be pressed for a long time and you want to save power a toggling or latching relay is best but they are bit more difficult to control.
simulate this circuit – Schematic created using CircuitLab
A transistor or FET may be possible but for that you would need the schematics and details of how the fan controller works.