I have two websites - One of them is Drupal 7, and the other is a different site (Site #2). In Site #2, within the user tables, is a field called "customid". Each user on Site #2 has a unique, randomly generated 30 character alphanumeric string within their respective "customid" field.

I can pull the value of the "customid" field for the user on Site #2, and use it as an input value in a form post to the Drupal 7 site.

In the Drupal 7 site, I created a custom user field "customid" as well. 

Is it possible to make a form post from Site #2 to Drupal 7 - with the "customid" value as an input, and if it matches the "customid" field in the Drupal user table, log them in, else - redirect to the registration page? 

We're essentially bypassing the username/password to log them in - I'm trying to log them in via custom user field.

I searched everywhere, and just couldn't find anything where something like this had been discussed. Thanks anyone for your time and help!

**UPDATE:**
<br><br>I've tried implementing W.M.'s solution below, however upon attempting to enable the module, I get the white screen of death. Nothing shows up in the logs for the error.

For my custom module, I have two files:<br>
ra_login.module<br>
ra_login.info

**Info file:**

    name = RA Login
    description = Check ID and Login if there is a match, else go to Registration Page.
    core = 7.x
    
    ; Information added by Drupal.org packaging script on 2013-12-16
    version = "7.x-1.0"
    core = "7.x"
    project = "ra_login"
    datestamp = "1387203506"

**Module File:**

    <?php
    
    function ra_login_menu()
    {
        $items['authenticate-by'] = array( // or any other custom path you choose
            'title' => 'Login by SSO',
            'access callback' => true, // no point in authenticating at this point in code execution
            'page callback' => 'ra_login_function',
            'type' => MENU_CALLBACK,
            );
        return $items;
    }
    
    function ra_login_function($CONTACTID)
    {
        // post the data to /authenticate-by as json with a parameter / property named CONTACTID holding the CONTACTID
        $postData = file_get_contents('php://input');
    
        $Data = json_decode($postData);
    
        $query = new EntityFieldQuery();
        $query->entityCondition('entity_type', 'user')
            ->propertyCondition('uid', 1, '<>')
            // User number 1 should not be able to log in using this method
            ->fieldCondition('field_contactid', 'value', $Data->CONTACTID, '=')
            ->range(0,1);
    
        $result = $query->execute();
    
        if (isset($result['user'])) {
            $uid = array_keys($result['user'])[0];
            $account = user_load($uid);
            global $user;
            $user->uid = $uid;
            $user->name = $account->name;
            $user->timezone = $account->timezone;
            user_login_finalize(); // Log user in
            drupal_goto('/user'); // or any other path to redirect logged in users to
        } else {
            drupal_goto('/user/register');
        }
    
    }

Again, for some reason when I try to enable the module, I get a white screen.
Thanks for anyone who can help!