Can I read a yaml file from a twig template?

You can from preprocess and set it into variables for the template.

If they're config yml you can get it directly from configuration, but if they're not you have to parse the yml file and get key/values from that, e.g. something along the lines of:

use Symfony\Component\Yaml\Parser;

$parser = new Parser();
$data = $parser->parse(file_get_contents($path_to_file));

Config is well covered in the docs, e.g. https://www.drupal.org/developing/api/8/configuration/simple