Ok, I finelly figured it out.
Looking at the code I noticed that in the back-end there was this script that weren’t in the front end so I added it in my form page and now the map is showing and the field is working fine :
I looked at google-map.php from the plugin code and it seems that the function input_admin_footer() which is calling acf/fields/google_map/api is used only for the back-end but not it the front-end, which caused my isue.
Welcome to the Advanced Custom Fields community forum.
Browse through ideas, snippets of code, questions and answers between fellow ACF users