Continuity across an LED. What kind of tester are you using if not a VOM? Even with a VOM it is sometime hard to see continuity across and diode. It just depends on the diode and on the meter. The general idea of a VOM to measure resistance or continuity is for the meter to apply a small current and measure the voltage across the terminals, OR to apply a small voltage and then measure the current that flows. In one direction (reverse mode) a diode conducts very little. In the other direction, forward bias, it conducts very little if the voltage is low. Once it gets to a higher voltage (0.5 volts for a Si diode or maybe 2 volts for an LED) then the conduction is higher and can be measured. What is being applied depends upon how the meter is designed.
First, the continuity test. The E-stop side should be a short when the button is not activated, NC. So if you push in the button then it should go to open circuit. Continuity or resistance test should work. You say continuity as oppose to resistance measurement so I am not for sure what kind of meter you might be using.
Second test. You say that there are two more terminals that you think are for the LED. If there is a + and - on them then great but if not don't worry. Try a 9 volt or 6 volt battery across the terminals that you think are the LED's. You are planning on hooking it up to 24 volt later so try a 9 volt now. If you have any resistors around put a 1000 Ohm, or so, in series with it. Throw the switch to test. It should light up with the switch in some position or all of the time, but it will be dimmer than with the 24 volt supply. If it does not light up reverse the polarity of the battery terminals and test again..
If you can give us some specs or a URL as to where you got it. Does it even say it is for 24 volt (AC or DC)?
Dave