What do I need to do to connect an ADC IC to a Adam 6066??Is it even possible to read the inputs from the ADC chip?
Cursory look at the data sheets (The ADAM datasheet leaves a little bit to be desired):  ADAM is max input frequency of 3KHz, ADC (MCP3208) is min output frequency of 10KHz.  Seems you will need a bridge to get the values from the ADC to the ADAM (reliably).  Maybe use a small CPLD to do this.

e.g.  ADC -> SPI Read of the ADC slave (GTE 10kHz) into a FIFO -> ADAM read/sample of ADC values from FIFO (LTE 3kHz)...  repeat.    

Not sure of the ADAM... those are 6 digital inputs.. so, maybe you have to send/apply the data to the ADAM and it samples it/them(?)  Bottom line, it does not appear that the ADAM is capable of reading the ADC directly.