Approximating dynamics in continuous-time with discrete-time
I'm wondering if the interactions and feedback will be able to
  accurately recreate the same type of chaotic behavior that truly
  continuous time inputs (such as those generated by a function
  generator) would produce.


Yes, it will be able to recreate the chaotic behavior, but only if the response of the non-linear system is limited to 100Hz (which it is probably not).

Since the input to your digital system is 100Hz that is limiter for the response of the digital system. 

If you can prove that the response of the nonlinear systems response is bounded to below 100Hz then you don't need to be worried about what the digital system is doing to the analog system.

One method that might help determine this is to insert one or more low pass filters in the nonlinear circuit and observing the result. One problem I foresee is the circuit may suffer from bandlimiting because nonlinear circuits change behavior from tiny fluctuations (butterfly effect). 

Another method that might help determine if the nonlinear circuit would suffer from limiting it to 100Hz would be to simulate this circuit in SPICE and change the the simulation time to a fixed time step with a rate of 100Hz

If the system works the DAQ will need a low pass filter on the output (probably at 100Hz) to recreate the signal and reduce the chopping from the zero order hold.


Source: RealHD audio

There are also different kinds of holds that you can use with a digital system to more accurately recreate a signal and the bi-linear transform  to determine the response:

 

Source: Slideplayer