Your code is almost right, just missed some details. To set the min and max axis value you need to use the viewWindow:{min: , max:} dictionary inside the hAxis and vAxis, as follows:

var chart2 =
    ui.Chart.image
        .series({
          imageCollection: lai_august_19_broadleaf,
          region: aoi,
          reducer: ee.Reducer.mean(),
          scale: 500,
          xProperty: "system:index"
        })
        chart2.setOptions({
          title: 'Mean LAI France broadleaf forest forest June-August 2019',
          hAxis: {
            title: 'Time',
            viewWindow: {min: 0, max: 6},
          },
          vAxis: {
            title: 'LAI',
            viewWindow: {min: 1, max: 7}
          },
        });

Your code is almost right, just missed some details. To set the min and max axis value you need to use the viewWindow:{min: , max:} dictionary inside the hAxis and vAxis, as follows:

var chart2 =
    ui.Chart.image
        .series({
          imageCollection: lai_august_19_broadleaf,
          region: aoi,
          reducer: ee.Reducer.mean(),
          scale: 500,
          xProperty: "system:index"
        })
        chart2.setOptions({
          title: 'Mean LAI France broadleaf forest forest June-August 2019',
          hAxis: {
            title: 'Time',
            viewWindow: {min: 0, max: 6},
          },
          vAxis: {
            title: 'LAI',
            viewWindow: {min: 1, max: 7}
          },
        });