Large language models (LLMs) have been widely deployed in Conversational AIs (CAIs), while exposing privacy and security threats. Recent research shows that LLM-based CAIs can be manipulated to extract private information from human users, posing serious security threats. However, the methods proposed in that study rely on a white-box setting that adversaries can directly modify the system prompt. This condition is unlikely to hold in real-world deployments. The limitation raises a critical question: can unprivileged attackers still induce such privacy risks in practical LLM-integrated applications? To address this question, we propose \textsc{VortexPIA}, a novel indirect prompt injection attack that induces privacy extraction in LLM-integrated applications under black-box settings. By injecting token-efficient data containing false memories, \textsc{VortexPIA} misleads LLMs to actively request private information in batches. Unlike prior methods, \textsc{VortexPIA} allows attackers to flexibly define multiple categories of sensitive data. We evaluate \textsc{VortexPIA} on six LLMs, covering both traditional and reasoning LLMs, across four benchmark datasets. The results show that \textsc{VortexPIA} significantly outperforms baselines and achieves state-of-the-art (SOTA) performance. It also demonstrates efficient privacy requests, reduced token consumption, and enhanced robustness against defense mechanisms. We further validate \textsc{VortexPIA} on multiple realistic open-source LLM-integrated applications, demonstrating its practical effectiveness.