Electronic – How to interface a pH probe

I don't really see a reason for this not to work the way you expect with the connections you have listed here. Might I suggest some debugging steps?

1) See what happens to the output range when you put a large-ish Rg ( say 27k ohms) in the circuit. Does the output swing change at all? 2) Have you checked the inputs to make sure they are as you expect with the amplifier connected? If so, how about using a larger resistance POT to simulate your resistor divider just to remove the sensor from the picture until you get the circuit working. 3) Perhaps the simplest explaination might be that the INA126 itself is bad. Do you have a spare you can drop in to test?

This should work, but usually there's also a resistor from the non-inverting input to ground:

If \$\dfrac{R4}{R2} = \dfrac{R3}{R1}\$ then:

\$V_{OUT} = \dfrac{R_{3}}{R_{1}} \times (V_{IN+} - V_{IN-})\$

To minimize offset error both inputs have to see the same impedance, and therefore \$R1 = R2\$ and \$R3 = R4\$.
Omitting R4 will only give a scaling factor for the voltage on the non-inverting input, namely:

\$V_{OUT} = 2 \times V_{IN+} - V_{IN-}\$

but changing the setting for the lower potmeter should definitely have a result in the output. Did you measure the voltages on both inputs?

What happens if you set the lower voltage to 2.5V and the upper to 1V? The inverting input should also be 2.5V, and the output 4V. What do you measure?

Note: especially the lower voltage follower is not necessary in your version; the opamp's input current is low enough to be negligible for most uses, and by the way, you're connecting the potmeter's wiper to an exactly same input!

Further reading
differential amplifier tutorial (interesting site overall!)