Hello from JustAnswer.
Yes, it is very possible that the armpit can get sore and tender from lifting your baby. The front and back of the armpit is composed of muscles that move the arm, so using the arm to lift a weight can cause a muscle strain. If there is an unusual position, then it is more likely that a strain can occur.
It is also true that there are many other conditions that can cause pain and soreness in the armpit, including several different infections or growths in the armpit.
If the soreness and tenderness started after lifting the baby, then that would be suspicious that it is a muscle strain. It also may help to try an anti-inflammatory medicine, and if the soreness and tenderness resolves, that would also support that this is due to a muscle strain.
If I can provide any additional information, please let me know.
Please let me know if I can provide any further assistance.