The word remains the same in French.

hth

That answer was my automatic answer, having heard the term used in English for a small musical instrument I used to play, as a child. However bilingual sources hinted to the term being also used for a glass organ or mouth organ. So I went to the English source and got:
"Harmonica: Small musical instrument. You play it by moving it accross your lips and by blowing and sucking air through it". I then went to the French source and got: "Harmonica: Petit instrument de musique rectangulaire dont le son est produit par des anches libres métalliques que l'on met en vibration en soufflant et en aspirant".
This confirms the term remains the same in both languages and that other possible translations are likely to refer to obsolete use of the term.

Hope this reassure you on the use of the same term in both language, which is something often causing problems.

This is the proper French word, as any French or English-French dictionary can confirm.

However, in French Canada, we also use the term "musique à bouche", which is not a very proper term, but everybody uses it and understands it.
We also use another term which is much more pretty and colourful. It is "un ruine-babines". "Babines" is a colloquial word meaning lips, as in the expression "le lécher les babines" (to lick one's chops). "Ruine-babines" of course, means something that "ruins one's chops or one's lips", which is what this instrument does to someone who plays too much of it.

Nevertheless, if you want to be proper and if you want any French speaker to understand you, you had better use the word "harmonica", which is more sedate, but to the point. :-)