How can I integrate a stemmer with Search API? Exactly what the question says - I'm using Search API with Database Search. I also have the Porter Stemmer module, but my understanding is that it only works with the Drupal built-in search.

Any help with this would be greatly appreciated.
It is possible to use the Porter Stemmer module with other modules besides the Drupal standard search. I have done it before (with Views exposed filters). 

In your module, you could either invoke the porterstemmer_search_preprocess() function or just copy its contents into your own function. And then pass your search terms to this function. E.g.:

function your_function_for_preprocessing_text($text) {
   // Include porterstemmer (change accordingly).
   module_load_include('module', 'porterstemmer');

  // Convert text to lower case, and replace special apostrophes with regular
  // apostrophes.
  $text = drupal_strtolower(str_replace('’', "'", $text));

  // Split into words
  $words = preg_split('/(' . PORTERSTEMMER_BOUNDARY . '+)/', $text, -1, PREG_SPLIT_DELIM_CAPTURE);

  if (!count( $words )) {
    return $text;
  }

  $has_pecl_stem = _porterstemmer_pecl_loaded();

  // Process each word, skipping delimiters.
  $isword = !preg_match('/' . PORTERSTEMMER_BOUNDARY . '/', $words[0] );
  foreach ($words as $k =&gt; $word) {
    if ($isword) {
      if ( $has_pecl_stem ) {
        $words[$k] = stem_english($word);
      }
      else {
        $words[$k] = porterstemmer_stem($word);
      }
    }
    $isword = !$isword;
  }

  // Put it all back together (note that delimiters are in $words).
  return implode('', $words);
}


As I said, I have used the Porter Stemmer to stem all the terms entered in a Views exposed search. However, you'll still need to find the right hook in the Search API module to pre-process the text before the search query is run against the database.