Refactor Snippet Edit File field to Vue

Problem to solve

Refactor File field on the Snippet edit form to Vue

Conflict with ACE-to-Monaco direction

This Issue conflicts with Replace ACE with Monaco for Snippets editing/creation in the way what should be implemented first. There are two possible scenarios so far:

1. This issue takes precedence. Then Replace ACE with Monaco for Snippets editing/creation should be implemented after this Issue and #207239 (closed) are closed.

2. Replace ACE with Monaco for Snippets editing/creation takes presedence. Then this Issue is blocked by #198604 (closed) and #198608 (closed)

Further details

Step 4 in the batch of "Refactor Snippets edit to Vue" works