Third-party libraries like Log4j accelerate software application development but introduce substantial risk. Vulnerabilities in these libraries have led to Software Supply Chain (SSC) attacks that compromised resources within the host system. These attacks benefit from current application permissions approaches: thirdparty libraries are implicitly trusted in the application runtime. An application runtime designed with Zero-Trust Architecture (ZTA) principles secure access to resources, continuous monitoring, and least-privilege enforcement could mitigate SSC attacks, as it would give zero implicit trust to these libraries. However, no individual security defense incorporates these principles at a low runtime cost. This paper proposes Zero-Trust Dependencies to mitigate SSC vulnerabilities: we apply the NIST ZTA to software applications. First, we assess the expected effectiveness and configuration cost of Zero-Trust Dependencies using a study of third-party software libraries and their vulnerabilities. Then, we present a system design, ZTD$_{SYS}$, that enables the application of Zero-Trust Dependencies to software applications and a prototype, ZTD$_{JAVA}$, for Java applications. Finally, with evaluations on recreated vulnerabilities and realistic applications, we show that ZTD$_{JAVA}$ can defend against prevalent vulnerability classes, introduces negligible cost, and is easy to configure and use.