Code review

Uploading module as a patch so I can comment easier without having to make an actual patch.

Overall this is looking very nice! Adding tests is of course a good idea, but you already have another issue open for this.
I did find some other things that can be changed, most things are coding standards related.

1. --- /dev/null
   +++ b/.gitignore
   

   This shouldn't be in the repo here. I don't see any other modules that have a gitignore.

2. +++ b/.gitignore
   --- /dev/null
   +++ b/README.txt
   

   Some line endings in this file have spaces. Those should be removed.

3. +++ b/README.txt
   @@ -0,0 +1,59 @@
   +It makes it possible to exclude nodes and other entities from being indexed in ¶
   +search indexed configured using Search API framework. ¶
   

   It makes it possible to exclude entities from being indexed in a Search API index.

4. +++ b/README.txt
   --- /dev/null
   +++ b/composer.json
   

   I don't think this is needed, the drupal composer facade already creates a composer file.

5. +++ b/search_api_exclude_entity.info.yml
   @@ -0,0 +1,8 @@
   +  - field
   +  - search_api
   

   I haven't seen any documentation about this, but as far as I know the best practice is to namespace dependencies. In this case that's drupal:field, search_api:search_api.

   This is not needed.

6. +++ b/search_api_exclude_entity.info.yml
   @@ -0,0 +1,8 @@
   \ No newline at end of file
   

   needs eof newline

7. +++ b/search_api_exclude_entity.module
   @@ -0,0 +1,34 @@
   \ No newline at end of file
   

   Needs eof newline.

8. +++ b/src/Plugin/Field/FieldFormatter/SearchApiExcludeEntityFormatter.php
   @@ -0,0 +1,44 @@
   +/**
   + * @file
   + * Contains Drupal\search_api_exclude_entity\Plugin\Field\FieldFormatter\SearchApiExcludeEntityFormatter.
   + */
   

   @file docblocks are not longer needed for files that contain only one class.

9. +++ b/src/Plugin/Field/FieldType/SearchApiExcludeEntityFieldItem.php
   @@ -0,0 +1,52 @@
   +/**
   + * @file
   + * Contains \Drupal\search_api_exclude_entity\Plugin\Field\FieldType\SearchApiExcludeEntityFieldItem.
   + */
   

   @file docblocks are not longer needed for files that contain only one class.

10. +++ b/src/Plugin/Field/FieldWidget/SearchApiExcludeEntityFieldWidget.php
    @@ -0,0 +1,82 @@
    +/**
    + * @file
    + * Contains \Drupal\search_api_exclude_entity\Plugin\Field\FieldWidget\SearchApiExcludeEntityFieldWidget.
    + */
    

    @file docblocks are not longer needed for files that contain only one class.

11. +++ b/src/Plugin/search_api/processor/SearchApiExcludeEntityProcessor.php
    @@ -0,0 +1,179 @@
    +/**
    + * @file
    + * Contains Drupal\search_api_exclude_entity\Plugin\search_api\processor\SearchApiExcludeEntityProcessor.
    + */
    

    @file docblocks are not longer needed for files that contain only one class.

12. +++ b/src/Plugin/search_api/processor/SearchApiExcludeEntityProcessor.php
    @@ -0,0 +1,179 @@
    +/**
    + * @SearchApiProcessor(
    

    I think this could use a docblock title.

13. +++ b/src/Plugin/search_api/processor/SearchApiExcludeEntityProcessor.php
    @@ -0,0 +1,179 @@
    +    $entity_manager = \Drupal::service('entity_field.manager');
    

    Proper DI here would be great, use ::create to inject the manager so we don't have to call \Drupal::service from here.

14. +++ b/src/Plugin/search_api/processor/SearchApiExcludeEntityProcessor.php
    @@ -0,0 +1,179 @@
    +      $entity_manager = \Drupal::service('entity_field.manager');
    

    and here.

15. +++ b/src/Plugin/search_api/processor/SearchApiExcludeEntityProcessor.php
    @@ -0,0 +1,179 @@
    +    if (isset($field_map[$entity_type][$field]['bundles'][$bundle])) {
    +      return TRUE;
    +    }
    +    else  {
    +      return FALSE;
    +    }
    

    return isset($field_map[$entity_type][$field]['bundles'][$bundle]) is the same thing.

16. +++ b/src/Plugin/search_api/processor/SearchApiExcludeEntityProcessor.php
    @@ -0,0 +1,179 @@
    +          // We need to be sure that the field actually exists
    +          // on the bundle before getting the value to avoid
    +          // InvalidArgumentException exceptions.
    

    This comment should be reflowed to be closer to 80 cols.

17. +++ b/src/Plugin/search_api/processor/SearchApiExcludeEntityProcessor.php
    @@ -0,0 +1,179 @@
    +            if (NULL !== $object->get($field)->getValue()[0]['value']) {
    

    The coding standards don't allow yoda if statements. The NULL should be on the other side of the !==