Disconnect Ethernet Shield
You have correctly determined the root of the problem, both shields are trying to use the same pins. To work around this we need to know which pins are being used, and for what. 

The Ethernet Shield documentation reveals the following:


  The Arduino board communicates with the shield using the SPI bus. This is on digital pins 11, 12, and 13 on the Uno and pins 50, 51, and 52 on the Mega. On both boards, pin 10 is used as SS. On the Mega, the hardware SS pin, 53, is not used to select the Ethernet controller chip, but it must be kept as an output or the SPI interface won't work.


https://www.arduino.cc/en/Reference/Ethernet

The documentation for Sparkfun's EasyVR library isn't as explicit, but one of the examples gives us another clue:

#if defined(__SAMD21G18A__)
  // Shield Jumper on HW (for Zero, use Programming Port)
  #define port SERIAL_PORT_HARDWARE
  #define pcSerial SERIAL_PORT_MONITOR
#elif defined(SERIAL_PORT_USBVIRTUAL)
  // Shield Jumper on HW (for Leonardo and Due, use Native Port)
  #define port SERIAL_PORT_HARDWARE
  #define pcSerial SERIAL_PORT_USBVIRTUAL
#else
  // Shield Jumper on SW (using pins 12/13 or 8/9 as RX/TX)
  #include "SoftwareSerial.h"
  SoftwareSerial port(12, 13);
  #define pcSerial SERIAL_PORT_MONITOR
#endif

#include "EasyVR.h"

EasyVR easyvr(port);


EasyVR-Arduino/examples/TestEasyVR/TestEasyVR.ino

This block of code is determining how to set up the serial communication. The first two if statements are for different boards, but the third one sets up the software serial connection... and it says there are two options, pins 12/13 OR 8/9! If there are two options, there must be a way to switch it on the board itself. 



And sure enough! Check out that white box labeled "SW Serial Pins". Jackpot!

Here's what the manual has to say:


  On the bottom side of the board there are two SMD resistors that you can move to select the two pins of
  Arduino that the EasyVR will be connected to when in Software Serial mode (Mode Jumper on SW).
  
  
  RX – Software Serial Receiver pin
  
  
  D12 – Use digital pin 12 as serial receiver (default)
  D8 – Use digital pin 8 as serial receiver
  
  TX – Software Serial Transmitter pin
  
  
  D13 – Use digital pin 13 as serial transmitter (default)
  D9 – Use digital pin 9 as serial transmitter
  
  
  
  The choice of pins 12-13 is maintained for backward compatibility with the previous hardware revisions of
  the EasyVR Shield. However those pins may also be used for the SPI interface, so another choice of pins 8-
  9 is provided. If you want to use different pins make sure the receiver pin supports change interrupts.