Basically, what you'll need to do is XML-ize your data and POST it to the API URLs they provide. I'll start out with the basic request, and we can build on the solution if you want.
$api_key = 'API_KEY'; // insert your API key here;
$ticket = 'TICKET_ID'; // your ticket here (should be with your account);
<FL val='Posting title'>$job_title</FL>
<FL val='Assigned recruiter'>$recruiter_id</FL>
<FL val='Job opening status'>$job_status</FL>
<FL val='Number of positions'>$num_positions</FL>
<FL val='Roles and responsibilities'>$job_description</FL>
You'd need to populate those variables with the values you've collected from the user. Then, you need to make a POST request. You can use the WP_Http class for that:
if( !class_exists( 'WP_Http' ) )
include_once( ABSPATH . WPINC. '/class-http.php' );
$url = "https://recruit.zoho.com/ats/private/xml/Module/addRecords?apikey=$api_key&ticket=$ticket";
$request = new WP_Http;
$result = $request->request( $url, array( 'method' => 'POST', 'body' => $data) );
Now, obviously you'll need to do some tweaking to add the data you'll need for your posting, and you'll need to handle the case that the request fails. But I hope this gives you a start (I can't guarantee the code will work 100%) - let me know and we can build on it!