Move "save settings" code from integration_settings_form.js to Vue component

What does this MR do and why?

This MR is part 4 of of #342121 (closed), an effort to remove app/assets/javascripts/integrations/integration_settings_form.js and move its current responsibilities to Vuex or existing Vue components.

• Part 1: !75484 (merged).
• Part 2: !75838 (merged)
• Part 3: !76291 (merged)

In this MR, we remove the SAVE_INTEGRATION_EVENT from the integration form event hub and related Vuex actions/mutations/state. We move all of the "save integration" functionality to integration_form.vue.

There are no user-facing changes in this MR.

Screenshots or screen recordings

These are strongly recommended to assist reviewers and reduce the time to merge your change.

How to set up and validate locally

1. Enable and configure the Jira integration for the project. More information.
2. Click save settings. The Test button should be disabled and the Save button should be loading while the request is in-flight. Once done, the page will refresh.
3. Play around with various states of the form (incomplete form, invalid form, etc.).

MR acceptance checklist

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• I have evaluated the MR acceptance checklist for this MR.

Related to #342121 (closed)