Have to hook into a function from a module I am using the LinkedIn Integration module and have created a custom module to add some functionality.

Please see the linkedin.inc file at line 17 - function linkedin_access_token($account)

When a user links their LinkedIn account, I want to add to the db_write_record to also write to a custom field in my database for a timestamp.

Not sure what hook to use since it is a function.
Just making a quickly research (I don't have time at this moment to test the code) inside the Linkedin Integration module I can tell you that I found a hook named linkedin_external_login

See the code in the linkedin_auth/linkedin_auth.module file in the line 124

/*
 * Implementation of hook_linkedin_external_login
 */

function linkedin_auth_linkedin_external_login($uid) {
  $name = db_query("SELECT authname FROM {authmap} WHERE uid = :uid AND module = :module", array(':uid' =&gt; $uid, ':module' =&gt; 'linkedin'))-&gt;fetchField();
  user_external_login_register($name, 'linkedin');
  drupal_goto();
}


Just implement your own hook_linkedin_external_login()