Select order type and store with hook_form_alter() In drupal commerce, I have 3 stores and 3 orders type
When placing a manual order, i am first asked to choose the store and the order type ( which are entities), with radio buttons. 
With a select list, I know how to programmatically select one option, with the following code.

function reservation_form_commerce_order_add_form_alter(&amp;$form, \Drupal\Core\Form\FormStateInterface $form_state, $form_id) {
  $form['type']['widget']['#default_value'][0] = 'type_order_1';
  $form['store_id']['widget']['#default_value'][0]['#value'] = 1;
}


How do I do the same with the button?
You're overthinking it. As with most elements, you just need to set #default_value on the main element definition itself. Here's working example code I tested in my own development environment:

/**
 * Implements hook_form_FORM_ID_alter().
 *
 * Sets default values on the order add form for the order type and store.
 */
function example_form_commerce_order_add_form_alter(&amp;$form, $form_state) {
  $form['type']['#default_value'] = 'default';
  $form['store_id']['#default_value'] = 2;
}


Obviously you'll need to replace the values I've set with your own. : )