24GHz Mixer - How to shift the frequency of a 24GHz(100MHz bandwidth) signal by +-4kHz
Note that this "GaAs MMIC I/Q MIXER" is, to put it bluntly, basically just a couple of fast diodes in a package !

"I will put the received signal to LO.... Does this have any chance of working?" Yes if your received signal is strong enough. Note that on the datasheet, first page above the table it says: LO = + 10 dBm
If your received signal is less than that, it might not work.

"Would it cause problems to put a 100MHz bandwidth signal to a LO port?"
Not as long as that 100 MHz is within the 21 - 30 GHz range.
On page 12 there's a schematic of the LO input, note that it is AC coupled and that there is an inductor. These are tuned to that 21 -30 GHz range so any signal outside that range will not be able to get in as the input is not tuned for that.

You mean that you want to frequency shift your received signal by 4 kHz by applying 4 kHz quadrature signals at the IF inputs, 24 GHz (BW = 100 MHz) at the LO input and then have the resulting 24.000004 GHz at the RF output ? That might work but I would start by applying the Received signal at the RF input instead if the LO. And even then I doubt whether the frequency shift will happen and if it does, if it will have enough conversion gain.

This module looks like a downconversion mixer to me so maybe it is not what you need.

Why don't you look for a module from for example MiniCircuits, Skyworks, Tryquint or Avago which is more suitable for this task ?