List all authors by matching custom meta data on a category page

Try the WP_User_Query, something like:

 $term = get_queried_object();

 $users = new WP_User_Query(array(
   'meta_key'   => 'your_meta_key',        // the key in which you store your terms
   'meta_value' => (int)$term->term_id,    // assuming you store term IDs
 ));

 print_r($users->get_results()); 

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A update based on your input:

$term = get_queried_object();

$user_query = new WP_User_Query(array(
  'role'         => 'subscriber',         // or whatever your target role is
  'meta_key'     => 'involvement', 
  'meta_compare' => 'like', 
  'meta_value'   => $term->name,          // or "%{$term->name}%" ?
  'fields'       => array('user_email'),  // if you only need the email
 )); 

$users = $user_query->get_results();

foreach($users as $user)
  echo get_avatar($user->user_email);

Just be aware that the “like” query can fail and return you the wrong user in case you have two terms with similar names, like “abc def” and “abc” – in the 2nd case you’ll get users with the “abc def” term too.

The only reliable way is to get all users with the meta data (remove meta arguments and change “fields” to “all_with_meta”), then check the meta for the exact term match, but this query is quite expensive…