According to French Wikipédia article:

The name is “La Crosse” (most of the time) and sometimes “Le Lacrosse“.

The sport has been named by the French Jean de Brébeuf as explained in the “Arrivée des Européens” part.

The selected answer contains the material related to the Indigenous peoples of the Americas origins for the game, even possibly dating back the 11th in Central America; and to Jean de Brébeuf coining “la crosse” around 1636 in the context of the First Nations of Canada.¹

The Wikipedia article from the answer supports “lacrosse” with a note based on a single reference, from a L’Équipe.fr article which contains for instance “Le Canada a créé une version indoor du lacrosse[…]”. Such an article may establish “lacrosse” regionally but has no direct bearing on the language as a whole. On the other hand the use can’t be rejected outright, since Larousse has it as a synonym for crosse. It is very likely the lacrosse use is supported in France because with one of the variants of the sports (au champ; field), the Association française de lacrosse has chosen “lacrosse en champ“; whereas both the Fédération de crosse du Québec et l’Association canadienne de crosse simply use “crosse” adding “au champ” to distinguish it from the indoor variation since it came out (crosse en enclos/salle; indoor). But in a North American context, even though it still appears as “vieilli” on Termium, I prefer a construction which, imho, fully leverages the French language, because after all this is where the word comes from:

L’emploi, en français, du terme anglais lacrosse est difficilement
justifiable. En effet, le terme lacrosse, qui représente un emprunt
ancien fait au français canadien (avec l’article la agglutiné), vient
doubler inutilement le terme bien établi crosse dont il est issu.
Ainsi, on dira : la crosse, et non : le lacrosse.

[ Grand dictionnaire terminologique – GDT “crosse” ]

This is consistent with casual use in situ. Therefore, in Québec and Canada: “(jeu de la)crosse, une partie de crosse, jouer à la crosse, un tournoi de crosse, j’ai perdu ma crosse” etc.

_{1. The etymology of lacrosse is pretty clear in English (Online E. Dict., AmHDotEL both refer to Canadian French “jeu de la crosse“). In the TLFi, you will find “crosse” (Crosse canadienne; jeu de la crosse; jouer à la crosse.); sure there is a reference to “cricket” and a Zola quote: “Au milieu de sa besogne, il [Zacharie] était resté sur le dos, les yeux vagues, rêvassant aux parties de crosse qu’il avait faites la veille” (Zola, Germinal, 1885, p. 1175). Was that about Lacrosse? Well no, it’s rather about the criquet, i.e. the crosse – that is the stick itself. In so many words, there is reference to a jeu de boules with a stick and therefore the reference can be construed as related to something akin to croquet/golf instead. But you will find crosse, crossage in context(compare with Zola) so it’s related in that sense. That’s all because Brébeuf might have coined the game he saw, but he didn’t invent the word “crosse“, which had been used since 1381 in the context of games.}

I just saw the question and feel like I can help

The sport itself is named “Lacrosse”.
When using it in sentences:

I play Lacrosse ==> Je joue au Lacrosse

It’s a Lacrosse stick ==> c’est une crosse de Lacrosse

Talking about the stick, we say “crosse” like in hockey.

Feel free to ask for more (I’m playing Lacrosse in France, created the Lille club and manage the Association Française de Lacrosse so I can be quite concerned :p)

You are trying to translate a French word into French. Lacrosse is French for “stick” or “the stick”. The penalty called “stepping on the crosse” literally means stepping on the stick.