Multi-label classification tasks such as OCR and multi-object recognition are a major focus of the growing machine learning as a service industry. While many multi-label prediction APIs are available, it is challenging for users to decide which API to use for their own data and budget, due to the heterogeneity in those APIs' price and performance. Recent work shows how to select from single-label prediction APIs. However the computation complexity of the previous approach is exponential in the number of labels and hence is not suitable for settings like OCR. In this work, we propose FrugalMCT, a principled framework that adaptively selects the APIs to use for different data in an online fashion while respecting user's budget. The API selection problem is cast as an integer linear program, which we show has a special structure that we leverage to develop an efficient online API selector with strong performance guarantees. We conduct systematic experiments using ML APIs from Google, Microsoft, Amazon, IBM, Tencent and other providers for tasks including multi-label image classification, scene text recognition and named entity recognition. Across diverse tasks, FrugalMCT can achieve over 90% cost reduction while matching the accuracy of the best single API, or up to 8% better accuracy while matching the best API's cost.