The way to alter the action handler for the search form is to add a submit handler, as outlined here: http://stackoverflow.com/questions/6444769/altering-search-form-action-not-working-for-custom-search-module-drupal-6. You can call your function in the submit handler or redirect to it from the submit handler.

/**
* Implementation of hook_form_alter()
*/
function mymodule_form_alter(&$form, &$form_state, $form_id) {
  if($form_id == 'search_block_form') {
    $form['#submit'][] = 'mymodule_form_submit';
  }
}

/**
 * Process a block search form submission.
 */
function mymodule_form_submit($form, &$form_state) {
  // The search form relies on control of the redirect destination for its
  // functionality, so we override any static destination set in the request,
  // for example by drupal_access_denied() or drupal_not_found()
  // (see http://drupal.org/node/292565).
  if (isset($_GET['destination'])) {
    unset($_GET['destination']);
  }

  // Check to see if the form was submitted empty.
  // If it is empty, display an error message.
  // (This method is used instead of setting #required to TRUE for this field
  // because that results in a confusing error message.  It would say a plain
  // "field is required" because the search keywords field has no title.
  // The error message would also complain about a missing #title field.)
  if ($form_state['values']['search_block_form'] == '') {
    form_set_error('keys', t('Please enter some keywords.'));
  }

  $form_id = $form['form_id']['#value'];
  $form_state['redirect'] = 'search/mymodule/'. trim($form_state['values'][$form_id]);
  }

The way to alter the action handler for the search form is to add a submit handler, as outlined here: https://stackoverflow.com/questions/6444769/altering-search-form-action-not-working-for-custom-search-module-drupal-6. You can call your function in the submit handler or redirect to it from the submit handler.

/**
* Implementation of hook_form_alter()
*/
function mymodule_form_alter(&$form, &$form_state, $form_id) {
  if($form_id == 'search_block_form') {
    $form['#submit'][] = 'mymodule_form_submit';
  }
}

/**
 * Process a block search form submission.
 */
function mymodule_form_submit($form, &$form_state) {
  // The search form relies on control of the redirect destination for its
  // functionality, so we override any static destination set in the request,
  // for example by drupal_access_denied() or drupal_not_found()
  // (see http://drupal.org/node/292565).
  if (isset($_GET['destination'])) {
    unset($_GET['destination']);
  }

  // Check to see if the form was submitted empty.
  // If it is empty, display an error message.
  // (This method is used instead of setting #required to TRUE for this field
  // because that results in a confusing error message.  It would say a plain
  // "field is required" because the search keywords field has no title.
  // The error message would also complain about a missing #title field.)
  if ($form_state['values']['search_block_form'] == '') {
    form_set_error('keys', t('Please enter some keywords.'));
  }

  $form_id = $form['form_id']['#value'];
  $form_state['redirect'] = 'search/mymodule/'. trim($form_state['values'][$form_id]);
  }