Vertical privacy-preserving machine learning (vPPML) enables multiple parties to train models on their vertically distributed datasets while keeping datasets private. In vPPML, it is critical to perform the secure dataset join, which aligns features corresponding to intersection IDs across datasets and forms a secret-shared and joint training dataset. However, existing methods for this step could be impractical due to: (1) they are insecure when they expose intersection IDs; or (2) they rely on a strong trust assumption requiring a non-colluding auxiliary server; or (3) they are limited to the two-party setting. This paper proposes IDCloak, the first practical secure multi-party dataset join framework for vPPML that keeps IDs private without a non-colluding auxiliary server. IDCloak consists of two protocols: (1) a circuit-based multi-party private set intersection protocol (cmPSI), which obtains secret-shared flags indicating intersection IDs via an optimized communication structure combining OKVS and OPRF; (2) a secure multi-party feature alignment protocol, which obtains the secret-shared and joint dataset using secret-shared flags, via our proposed efficient secure shuffle protocol. Experiments show that: (1) compared to the state-of-the-art secure two-party dataset join framework (iPrivjoin), IDCloak demonstrates higher efficiency in the two-party setting and comparable performance when the party number increases; (2) compared to the state-of-the-art cmPSI protocol under honest majority, our proposed cmPSI protocol provides a stronger security guarantee (dishonest majority) while improving efficiency by up to $7.78\times$ in time and $8.73\times$ in communication sizes; (3) our proposed secure shuffle protocol outperforms the state-of-the-art shuffle protocol by up to $138.34\times$ in time and $132.13\times$ in communication sizes.