Enable application submission to backend route

[samderanova]

Currently, submitting an application does not properly send it to the backend. The form's action should be set to /api/user/apply.

[taesungh]

As noted in #97, the native POST request will work for now, but it would be better to have client-side error handling in case anything goes wrong during the form submission (notably user session expires) in which case we would need to use event.preventDefault and then send the POST request ourselves with axios or alike.

[taesungh]

Thought we unlinked it from the PR but guess GitHub thought otherwise.
Additional note looking from last year's: the application form contains some write-in "Other" fields which we previously handled with the client-side submission. Since we are using SSR, it may be worth implementing this on the API side since someone could technically have JavaScript disabled. We can revisit this.

[samderanova]

To better handle situations in which a user's session has expired, we should write some additional code to send a request using axios along with error handling to display a notification when the session expires.

As a result, we also need to handle the situation where a user has JavaScript disabled. The axios request would not be sent, so we automatically fall back on native POST requests. If a user has JS disabled and submits without selecting the "Other" radio button for any question that contains such a button, then we can safely continue to process this request in the backend. Otherwise, we should display some 400 status code (see #93) to tell the user to enable JS because the native POST request will send "Other" as the value for the specific question (i.e. major, pronouns, etc.) instead of the actual value the user provided. Note that the user would be seeing this as a raw API response because our page would not be able to handle the response.