OPA2376: How much current can the OPA2376 source and sink?

Joe,

Op Amp Input and Output Swing Limitations cover this topic in detail.  Section 13 covers the Short-Circuit limit and section 10 covers Output Voltage Swing vs Output Current.  In summary, you want to look at the Output Voltage vs Output Current curve and make sure that the output voltage is below the curve (linear operating area).  For best accuracy, you want to give some margin in your selected safe operating area.  

See the image below for OPA376 Output voltage vs Output current.  In this case I am considering the temperature to be 25C.  For that temperature we could get a 2V linear output signal at 20mA with some margin (shown in green).  Using the same graph, notice that you cannot get much above 2.2V on a 2.75V power supply (see red "x").  Also, notice that at 125C the output voltage vs output current curve is to the left (curve shown in red).  For 125C the output current could not be much beyond 10mA to achieve 2V of output swing.

Thus, the answer to your question really depends on the temperature you are operating at as well as your output voltage requirement.  From a safe operation perspective teh short-circuit protection will prevent damage to the device, but if you want linear operation you need to stay inside the Output Swing curve (in the linear operating area).  Finally, as you approach the saturation curve the amplifier output will start to become distorted.  Thus, you want to add margin to the design by keeping below the curve by a few hundred millivolts.  

I hope this helps, Art