Recommender systems typically represent users and items by learning their embeddings, which are usually set to uniform dimensions and dominate the model parameters. However, real-world recommender systems often operate in streaming recommendation scenarios, where the number of users and items continues to grow, leading to substantial storage resource consumption for these embeddings. Although a few methods attempt to mitigate this by employing embedding size search strategies to assign different embedding dimensions in streaming recommendations, they assume that the embedding size grows with the frequency of users/items, which eventually still exceeds the predefined memory budget over time. To address this issue, this paper proposes to learn Scalable Lightweight Embeddings for streaming recommendation, called SCALL, which can adaptively adjust the embedding sizes of users/items within a given memory budget over time. Specifically, we propose to sample embedding sizes from a probabilistic distribution, with the guarantee to meet any predefined memory budget. By fixing the memory budget, the proposed embedding size sampling strategy can increase and decrease the embedding sizes in accordance to the frequency of the corresponding users or items. Furthermore, we develop a reinforcement learning-based search paradigm that models each state with mean pooling to keep the length of the state vectors fixed, invariant to the changing number of users and items. As a result, the proposed method can provide embedding sizes to unseen users and items. Comprehensive empirical evaluations on two public datasets affirm the advantageous effectiveness of our proposed method.