Which is the right thing to say?
- Put the thermometer in the armpit.
- Put the thermometer under the armpit.
- Put the thermometer under the arm.
The word armpit is defined as "a hollow under the arm at the shoulder" and therefore I would go with option 1: "Put the thermometer in the armpit." as I strongly believe we put something in a hollow rather than under it, though the expression "under the armpit" is still quite common, probably because many people visualize armpit as "the joint where the arm connects to the shoulder", and thus I don't consider "under the armpit" ungrammatical but technically speaking we don't put things under a hollow.
The expression "under the arm" is grammatical but it describes a less precise location as the arm is defined to be "the part of the upper limb between the shoulder and the elbow joints". Nevertheless, I think people would know which region "under the arm" you are talking about when thermometer is mentioned.