How do I hijack a reading from the circuit of my digital meter
You have received some excellent answers on intercepting the data in the digital domain but it may also be worth considering using the analog signals presented at pins 30 and 31 on the ICL7136. An adantage to this is only having a pair of lines to connect per device and probably a simpler software interface. Some things you'd need to check are:


Make sure attaching a small piece of shielded wire and the ADC channel doesn't interfere with the accuracy of the existing readings.
Check that the output voltage swing is sufficient to give enough resolution for your application with the AVR's ADC.


Possible solutions to a lack of resolution would be to add some extra signal conditioning and/or use a higher resolution external ADC. It's another avenue worth investigating depending on your requirements.