Algorithm audits are powerful tools for studying black-box systems. While very effective in examining technical components, the method stops short of a sociotechnical frame, which would also consider users as an integral and dynamic part of the system. Addressing this gap, we propose the concept of sociotechnical auditing: auditing methods that evaluate algorithmic systems at the sociotechnical level, focusing on the interplay between algorithms and users as each impacts the other. Just as algorithm audits probe an algorithm with varied inputs and observe outputs, a sociotechnical audit (STA) additionally probes users, exposing them to different algorithmic behavior and measuring resulting attitudes and behaviors. To instantiate this method, we develop Intervenr, a platform for conducting browser-based, longitudinal sociotechnical audits with consenting, compensated participants. Intervenr investigates the algorithmic content users encounter online and coordinates systematic client-side interventions to understand how users change in response. As a case study, we deploy Intervenr in a two-week sociotechnical audit of online advertising (N=244) to investigate the central premise that personalized ad targeting is more effective on users. In the first week, we collect all browser ads delivered to users, and in the second, we deploy an ablation-style intervention that disrupts normal targeting by randomly pairing participants and swapping all their ads. We collect user-oriented metrics (self-reported ad interest and feeling of representation) and advertiser-oriented metrics (ad views, clicks, and recognition) throughout, along with a total of over 500,000 ads. Our STA finds that targeted ads indeed perform better with users, but also that users begin to acclimate to different ads in only a week, casting doubt on the primacy of personalized ad targeting given the impact of repeated exposure.