### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

APiGateway Model that being created by CDK project stores the JSON schema with a single quotes instead of double quotes which leading to an error in parsing this json object. 

you can refere to this article to see the steps 

https://dev.to/dvddpl/reduce-lambda-invocations-and-code-boilerplate-by-adding-a-json-schema-validator-to-your-api-gateway-15af

and it works fine on AWS account 

also you can look on the attached image how localstack stores the JSON object with a single quotes, I tried to edit it manually through UI but it return it back with single quotes

![Screen Shot 2022-10-18 at 4 28 08 PM](https://user-images.githubusercontent.com/106184681/196769273-0834865a-2fb3-4efc-b3a9-9da6b2ad511d.png)

### Expected Behavior

store the json schema in correct JSON format

### How are you starting LocalStack?

With the `localstack` script

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

    localstack start

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

    cdklocal deploy


### Environment

```markdown
- OS:OSX
- LocalStack: 1.0.4
```


### Anything else?

_No response_
self.router = router or ApigatewayRouter(ROUTER)
    def on_after_init(self):
        apply_patches()
        self.router.register_routes()
    @handler("TestInvokeMethod", expand=False)
    def test_invoke_method(
        self, context: RequestContext, request: TestInvokeMethodRequest
    ) -> TestInvokeMethodResponse:
        invocation_context = to_invocation_context(context.request)
        invocation_context.method = request.get("httpMethod")
        invocation_context.api_id = request.get("restApiId")
        invocation_context.path_with_query_string = request.get("pathWithQueryString")
        moto_rest_api = get_moto_rest_api(context=context, rest_api_id=invocation_context.api_id)
        resource = moto_rest_api.resources.get(request["resourceId"])
        if not resource:
            raise NotFoundException("Invalid Resource identifier specified")
        invocation_context.resource = {"id": resource.id}
        invocation_context.resource_path = resource.path_part
        if data := parse_json_or_yaml(to_str(invocation_context.data or b"")):