Offers a new "JSON Data" field when you "Add fields" in the Drupal search_api interface. If you set the field to "Full text" it will be searchable.

JsonapiData.php

<?php

namespace Drupal\PROJECT\Plugin\search_api\processor;

use Drupal\search_api\Datasource\DatasourceInterface;
use Drupal\search_api\Item\ItemInterface;
use Drupal\search_api\Processor\ProcessorPluginBase;
use Drupal\search_api\Processor\ProcessorProperty;

/**
 * Adds the item's URL to the indexed data.
 *
 * @SearchApiProcessor(
 *   id = "jsonapi_data",
 *   label = @Translation("{jsonapi} data"),
 *   description = @Translation("Data as it appears on a JSON API endpoint"),
 *   stages = {
 *     "add_properties" = 0,
 *   },
 *   locked = true,
 *   hidden = true,
 * )
 */
class JsonapiData extends ProcessorPluginBase {

  /**
   * {@inheritdoc}
   */
  public function getPropertyDefinitions(DatasourceInterface $datasource = NULL) {
    $properties = [];

    if (!$datasource) {
      $definition = [
        'label' => $this->t('{jsonapi} data'),
        'description' => $this->t('Data as it appears on a JSON API endpoint'),
        'type' => 'string',
        'processor_id' => $this->getPluginId(),
      ];
      $properties['jsonapi_data'] = new ProcessorProperty($definition);
    }

    return $properties;
  }

  /**
   * {@inheritdoc}
   */
  public function addFieldValues(ItemInterface $item) {
    $entity_type_id = $item->getDatasource()->getEntityTypeId($item->getOriginalObject());
    $id = $item->getOriginalObject()->getValue()->id();

    $entityManager = \Drupal::service('entity_type.manager')->getStorage($entity_type_id);
    $entityToJsonApi = \Drupal::service('jsonapi.entity.to_jsonapi');
    $entity = $entityManager->load($id);
    $serialized_entity = $entityToJsonApi->serialize($entity);

    $fields = $this->getFieldsHelper()
      ->filterForPropertyPath($item->getFields(), NULL, 'jsonapi_data');
    foreach ($fields as $field) {
      $field->addValue($serialized_entity);
    }
  }

}

Sample endpoint code

<?php

// I'm not sure how default search might handle it, but in a custom endpoint
// it looks something like this (untested) code.

$index = SearchIndex::load('acquia_search_index');
$query = $index->query($options);
$query->keys("some search string");
$results = $query->setSearchId($search_id)->execute();

/** @var \Drupal\search_api\Item\ItemInterface $result */
foreach ($results->getResultItems() as $item_id => $result) {
  if ($fieldItem = $result->getField('jsonapi_data')) {
    $values = $fieldItem->getValues();
    $some_response_data['some_key'] = json_decode($values[0], TRUE)['data'];
  }
}