Well, if "connect the other side of the LED to an analog pin to read the value" means you're going to use a LED as a diode in the signal path, then I'd say the odds are slim it'll work in any useful way. Even regular diodes are problematic, due to their forward voltage drop. LEDs have far greater forward voltage, and some have terribly slow specs like reverse recovery time, which would probably cause all sorts of other trouble.
An analog circuit which tracks the envelope of a signal would normally use a
opamp-based precision rectifier circuit driving a high quality capacitor (eg, a NP0 or plastic film type), plus either a resistor to drain the capacitor slowly, or a switch to reset the measurement.
Of course, you're probably looking to track the envelope of the LFO, not the actual PWM waveform, so before you can even try to feed it into a diode+capacitor circuit to capture the signal's envelope, you'll need to low pass filter the PWM to get the LFO waveform. If you look at the
waveforms on the PWM page, you can see the trade-offs in speed versus the smoothness of the filtered output. Obviously any saw-tooth like ripple left on the signal is going to play havoc with the results from a diode-based envelope follower circuit.
If you over come all these analog circuit design issues, a diode-based signal amplitude circuit is essentially a peak detector, which is the same as the audio library's peak detect, except with analog noise and distortion added. Presumably you wish to go to all this extra effort for some reason... which I don't understand, but whatever you're hoping to accomplish, you'll need to design your analog circuit to do.
Teensy 3.1 has 12 PWM pins. If you're controlling them with analogWrite, you could use all 12. If you're using the audio shield, 3 of those 12 pins are required to communicate with the SGTL5000 chip, leaving 9 available. If you're using the SD card portion, one more of those pins is needed to control the SD card.
If you use the audio library to automatically send the signal to PWM, only a single signal is supported.