uses
  System.Permissions,
{$IFDEF ANDROID}
  Androidapi.Jni.Os,
  Androidapi.Helpers,
{$ENDIF}
...
  FPermissionCoarseLocation: string;
  FPermissionFineLocation: string;
...
procedure TMainForm.FormCreate(Sender: TObject);
begin
{$IFDEF ANDROID}
  // Note we can alternatively use literal strings:
  //   'android.permission.ACCESS_COARSE_LOCATION'
  //   'android.permission.ACCESS_FINE_LOCATION'
  // according to the documentation:
  //   https://developer.android.com/reference/android/Manifest.permission#ACCESS_COARSE_LOCATION
  //   https://developer.android.com/reference/android/Manifest.permission#ACCESS_FINE_LOCATION
  FPermissionCoarseLocation := JStringToString(TJManifest_permission.JavaClass.ACCESS_COARSE_LOCATION);
  FPermissionFineLocation := JStringToString(TJManifest_permission.JavaClass.ACCESS_FINE_LOCATION);
{$ENDIF}
end;

procedure TMainForm.Button1Click(Sender: TObject);
begin
  PermissionsService.RequestPermissions([FPermissionCoarseLocation, FPermissionFineLocation], LocationPermissionRequestResult);
end;