feat(web-ide): Use auth provider if available

• Context MR: Draft: wip(web-ide): Use DefaultApiClient when ... (!1402 - closed)
• Part 1: refactor: Move DefaultApiClient out of GitLabSe... (!1418 - merged)
• Part 2: feat(web-ide): Use auth provider if available (!1426 - merged)
• Part 3 (in Web IDE): feat: Update vscode extension to use auth provider (gitlab-web-ide!288 - merged)

Description

Related Issues

#1253 (closed)

Resolves #[issue_number]

How has this been tested?

Tested over in feat: Update vscode extension to use auth provider (gitlab-web-ide!288 - merged) by pointing gitlab-vscode-extension submodule to this branch. Using OAuth token in here works 👍

Screenshots (if appropriate)