Drupal Form API, add an image field that includes a preview and an alt text field to your custom module's configuration form.

SettingsForm.php

<?php
// src/Form/SettingsForm.php
/* Include theses dependencies in my_custom_module.info.yml
dependencies:
  - drupal:file
  - drupal:image
*/

namespace Drupal\my_custom_module\Form;

use Drupal\Core\Form\ConfigFormBase;
use Drupal\Core\Form\FormStateInterface;
use Drupal\Core\Security\TrustedCallbackInterface;
use Drupal\file\Element\ManagedFile;
use Drupal\file\Entity\File;

/**
 * Configure My Custom Module's settings for this site.
 */
class SettingsForm extends ConfigFormBase implements TrustedCallbackInterface {

  /**
   * {@inheritdoc}
   */
  public function getFormId() {
    return 'my_custom_module_settings';
  }

  /**
   * {@inheritdoc}
   */
  protected function getEditableConfigNames() {
    return ['my_custom_module.settings'];
  }

  /**
   * {@inheritdoc}
   */
  public function buildForm(array $form, FormStateInterface $form_state) {
    $config = $this->config('my_custom_module.settings');

    $form['general'] = [
      '#type' => 'details',
      '#title' => $this->t('Sitewide settings'),
      '#collapsible' => TRUE,
      '#open' => TRUE,
    ];
    // Add a managed file form element for uploading an image.
    $form['general']['my_custom_module_image'] = [
      '#type' => 'managed_file',
      '#title' => t('Image'),
      '#description_display' => 'before',
      '#upload_location' => 'public://',
      '#upload_validators' => [
        'file_validate_extensions' => ['gif png jpg jpeg'],
        'file_validate_is_image' => [],
      ],
      '#process' => ['::processImage'],
      '#theme' => 'image_widget',
      '#default_value' => $config->get('my_custom_module_image'),
    ];
    $form['general']['my_custom_module_image']['alt'] = [
      '#type' => 'textfield',
      '#title' => t('Alternative text'),
      '#maxlength' => 512,
      // If we don't bust this element out of the tree, it doesn't appear at all
      // in $form_state after submission. I cannot figure out why.
      '#tree' => FALSE,
      '#default_value' => $config->get('my_custom_module_image_alt'),
    ];

    return parent::buildForm($form, $form_state);
  }

  /**
   * {@inheritdoc}
   */
  public function submitForm(array &$form, FormStateInterface $form_state) {

    // First, set image file to permanent so Drupal doesn't try to clean it up.
    if (!empty($form_state->getValue('my_custom_module_image'))) {
      $fid = reset($form_state->getValue('my_custom_module_image'));
      $file = File::load($fid);
      $file->setPermanent();
      $file->save();
    }

    $this->config('my_custom_module.settings')
      ->set('my_custom_module_image', $form_state->getValue('my_custom_module_image'))
      ->set('my_custom_module_image_alt', $form_state->getValue('alt'))
      ->save();
    parent::submitForm($form, $form_state);
  }

  /**
   * {@inheritdoc}
   */
  public static function trustedCallbacks() {
    return [
      'processImage',
      'preRenderImage',
    ];
  }

  /**
   * Tweak the Image field with a #process callback.
   */
  public function processImage($element, &$form_state, $form) {
    $element = ManagedFile::processManagedFile($element, $form_state, $form);

    // If we remove the image, make sure the custom alt text goes away too.
    if (empty($element['#value']['fids'])) {
      $element['alt']['#default_value'] = NULL;
    }

    $element['#pre_render'][] = [SettingsForm::class, 'preRenderImage'];
    return $element;
  }

  /**
   * If an image is uploaded, show an image preview and an alt text field.
   */
  public static function preRenderImage(array $element) {

    // If a custom image's been uploaded.
    if (!empty($element['#value']['fids'])) {

      // There's only ever going to be one file here.
      $fid = reset($element['#value']['fids']);
      $file = File::load($fid);

      $image = \Drupal::service('image.factory')->get($file->getFileUri());

      // This will add preview of image at full size.
      // If your image is not small you likely want to use
      // '#theme' => 'image_style',
      // '#style_name' => 'thumbnail', // choose an existing style.
      $element['preview'] = [
        '#theme' => 'image',
        '#uri' => $image->getSource(),
        '#alt' => $file->getFilename(),
        '#height' => $image->getHeight(),
        '#width' => $image->getWidth(),
      ];
    }
    else {
      // If there's no file uploaded, hide the Alt text field.
      $element['alt'] = [
        '#access' => FALSE,
      ];
    }
    return $element;
  }

}