Transmitters
No, there are more than 850, 880, and 940nm transmitters: the IR spectrum ranges from 700nm to 1mm. A distinct set of values is sold, typically in the 700 to 1400 nm range (IR-A), where 850, 880, and 940 are common values. Here's a selection of the Mouser listing of IR emitters. I've used the TSAL series of emitters from Vishay, although I'm not sure that they're in your current, brightness, or bandwidth specs.
The parameter you're interested in for transmitters is the "Relative Radiant Power vs. Wavelength", and for receivers, you want to know the "Relative Spectral Sensitivity vs. Wavelength". For instance, the bell curve in Fig. 9 of the datasheet for the TSAL6100 shows that it has a relative intensity of 1 at 940nm, and outputs about 0.125 times this intensity at 890 nm. That likely means that it's not bright enough to use with an 850 nm detector, and would be iffy at best with an 880nm detector.
Receivers
On the plus side, receivers are usually more generous, for example the TSOP348 detector [picked at random] has a spectral sensitivity of better than 80% for all wavelengths between 850 and 1050nm. Taos Inc. also makes some nice digital, analog, and frequency output detectors for many wavelengths; I've used them with good success before. This will help you if you need to replace a sensor, especially if it's just used as an on/off digital sensor, for instance in a light curtain application, because 80% is pretty close to 100%.
However, that sort of receiver will only tell you about the quantity of light. If you knew that your LEDs were the same brightness (you don't), then you might be able to infer a frequency (i.e. this one is 75% as bright as the 950nm, therefore it's about 820 or 1070nm). You can also determine that an LED is on with just a digital camera, like the one in your cell phone.
Color Sensors
An infrared camera could tell you the wavelength after compensating for temperature, but would not fit most budgets. (Note: These are awesome for determining all kinds of things - Night vision, temperature gradients, etc.)
What you need for that is an color sensor in the infrared range. A color sensor will have multiple narrow-band and/or filtered detectors, so that you can determine the color of the light. See Figure 1, Photodiode Spectral Responsivity" of the datasheet for the TAOS TCS3200D[pdf] for an example (No, it's not going to be a pretty algorithm...). However, you'll notice that the visible light filters stop at about 750nm, and everything goes back to the same curve. Finding a color sensor that works into the infrared range is left as an exercise to the reader, but this sort of IC is what you're looking for.
An alternative to an IR color sensor (which may not exist) would be to use a broadband sensor with a set of infrared transmitting filters tuned to the region of the spectrum which you need. A quick Google search turned up this page, you'll probably find something better.
Distributors:
As for distributors, I find that Mouser has a better selection and cheaper prices on optoelectronics than Digikey.
Why LED2 doesn't turn on:
You should be aware that the LM324 series of op-amps has not got the output stage to source or sink significant current at only 2V margins. If that is not the problem causing your LEDs only very slightly lighting up, it may well be that your finger creates a 200kOhm or even higher resistance, depending on the distance between plates, the dryness of your skin and amount of pressure.
If the problem is not the output current capabilities of the op-amp in such a low voltage set-up, it may well be your 10kOhm resistor is pulling down too strongly for your finger to make sufficient difference to the op-amp. Normally that should not be a problem, as it should normally still at least reach 80mV or so, but you could try and increase that resistor to 100kOhm, to create a stronger offset in the op-amp's initial stage, in turn forcing the output slightly further into saturation.
Why LED1 doesn't turn on:
The op-amp's input terminals are both at ground level when nothing is touched, this means there is no difference between the terminals and the output stage of the op-amp is very lowly motivated to do something. If you want the output to come close enough to sinking any current at all, you need to bias the other input to a voltage above 0V.
My suggestion:
simulate this circuit – Schematic created using CircuitLab
Why the two resistors on the LEDs?: If your op-amp enters an undetermined state because the V+ and V- come close to each other, the output may start to float. In this case the LEDs could force the set-point, by conducting directly through each-other. Without any resistors in the current carrying path. This would be bad. By adding the two resistors, in this transitionary state the LEDs will never go past their current limit.
Why LM358?: Because it has a stronger output stage.
The rest has been explained above.
If it were not 4:30AM here I might add another schmidt-trigger tutorial here to make sure that transitionary state never occurs long enough to be a problem, but this time, I will leave the concept of an Op-Amp based Schmidt Trigger as further autonomous research.
Best Answer
So you're seeing about 1.5v on the output, but the LM324N datasheet says that the output can drive down to at least 20 mV (into a 10K load). It also seems unlikely that something is pulling the output high(er). This would imply that the problem is on the input side of things.
The datasheets says that the inputs can go to 0V, but I suspect that this is not entirely true. It might be that they only go to "almost" 0v. Even a couple of mV higher than 0v might be enough to mess up your circuit.
I suggest changing your circuit. It's a rather major change, unfortunately. The easiest (from a testing standpoint) would be to switch to a different opamp that can run off of a +/- power rail. If the opamp has a negative rail then it will be fine if the input is close to 0v (and not V-). You might be able to run the existing opamp that way, but the datasheets are ambiguous about that (and I'm too tired to look into it closely).
A different change, which would be much more complex, unfortunately, would be to change the circuit so that the input to the opamp never gets close to 0v. There are several ways to do this, but none of them are easy. One way is to put the current sense resistors on the high side, between the power rail and your LED's. Then use a resistor-divider to bring that voltage down within the range of the opamp. This might require changing some polarities and such (swapping the + and - inputs), but don't take my word for that-- actually figure it out first.
Another possibility where this circuit is going wrong is that it is actually unstable. You might need a cap between the opamp output and the negative input, and a resistor between the transistors and the negative input. I usually figure that stuff out in a simulator, but if you wanted to guess at some values I would go with 10K resistor and a 100 pF cap.