I need a circuit to turn on a LED with 0.2 volts. I want to do it with a transistor as a switch. What kind of transistor will turn on at this voltage?
No single transistor can do what you want. Instead, you need to use multiple transistors. A simple voltage comparator will do what you want:
R2 and R3 set the reference voltage for the base of Q2. As long as the voltage at the base of Q1 is less than Vref, all of the current through R1 will flow through Q2. As soon as the input voltage rises above 0.2V, Q2 will cut off and all of the current will flow through Q1 and the LED instead.
Note that since the emitters of the transistors need to be at a negative voltage, this circuit requires a negative supply. An alternative would be to use an IC comparator — you can find single-supply units that can work with input voltages close to ground.
Whilst I'm inclined to agree with @Bimpelrekkie that this is probably an X-Y problem, for those who might actually need such a device there are some options. Most MOSFETs are sold as enhancement or depletion mode (less commonly) devices, however it's possible to tune the threshold voltage to approximately zero +/- tens of mV.
For example, the Advanced Linear Devices ALD110800A and similar devices in that series have a nominal zero gate threshold (1uA Ids). The subthreshold behavior is fairly well documented as well.
For your actual situation, use CrossRoads circuit with just about any single supply op-amp or single-supply comparator. LM358 or LM393 will work fine from a 5V supply.
You could also use an Op Amp, or a Comparator. Set one input to 0.2V, when the other input exceeds that, the output will switch.
Here's one example
A well chosen germanium transistor can do this. I have some that switch at .15V. Finding one might be a problem ; I am not offering any of mine, but they still exist. If you need accurateness use one of the other answers, The simplest solution is to prebias a transistor so the base receives enough voltage to turn on when the input is .2V. I have used this trick in non-critical situations.