I have a Vigier with a non-locking vibrato and my guitar goes out of tune when I use the vibrato.
– You must check that the nut does not jam the strings, there are 3 types of nut
1 dot= gauge 9-42 to 9-46
2 dots= gauge 10-52
3 dots= gauge 12-56
You must choose a nut adapted to your string gauge.

– Check that you do not wrap the strings around the tuners shaft, see the manual for how to do this.

– Check that you have not loosened the screw at the end of the machine head button.

– You can also put a spot of grease under the part of the string that rests on the zero fret.

– Check that the Flexretainer is vertical