How can I get my WordPress plugin to receive data and relay it in an ajax/php request to a remote server that requires authentication?

In your PHP to call the remote server you may want to look at the http and API functions availalble in WordPress here: https://developer.wordpress.org/plugins/http-api/

For example, here’s a much easier way to get data from a remote server:

$response = wp_remote_get( 'https://api.github.com/users/blobaugh' );
$http_code = wp_remote_retrieve_response_code( $response );

And here’s a way to post like you were submitting a form:

$body = array(
    'sendThisValue'    => 'value',
);

$args = array(
    'body'        => $body,
// see docs for rest of parameters here
);

$response = wp_remote_post( 'http://your-contact-form.com', $args );