Continual learning (CL) aims to learn new tasks without forgetting previous tasks. However, existing CL methods require a large amount of raw data, which is often unavailable due to copyright considerations and privacy risks. Instead, stakeholders usually release pre-trained machine learning models as a service (MLaaS), which users can access via APIs. This paper considers two practical-yet-novel CL settings: data-efficient CL (DECL-APIs) and data-free CL (DFCL-APIs), which achieve CL from a stream of APIs with partial or no raw data. Performing CL under these two new settings faces several challenges: unavailable full raw data, unknown model parameters, heterogeneous models of arbitrary architecture and scale, and catastrophic forgetting of previous APIs. To overcome these issues, we propose a novel data-free cooperative continual distillation learning framework that distills knowledge from a stream of APIs into a CL model by generating pseudo data, just by querying APIs. Specifically, our framework includes two cooperative generators and one CL model, forming their training as an adversarial game. We first use the CL model and the current API as fixed discriminators to train generators via a derivative-free method. Generators adversarially generate hard and diverse synthetic data to maximize the response gap between the CL model and the API. Next, we train the CL model by minimizing the gap between the responses of the CL model and the black-box API on synthetic data, to transfer the API's knowledge to the CL model. Furthermore, we propose a new regularization term based on network similarity to prevent catastrophic forgetting of previous APIs. Our method performs comparably to classic CL with full raw data on the MNIST and SVHN datasets in the DFCL-APIs setting. In the DECL-APIs setting, our method achieves 0.97×, 0.75× and 0.69× performance of classic CL on the more challenging CIFAR10, CIFAR100, and MiniImageNet, respectively.