This paper introduces PoSSUM, an open-source protocol for unobtrusive polling of social-media users via multimodal Large Language Models (LLMs). PoSSUM leverages users' real-time posts, images, and other digital traces to create silicon samples that capture information not present in the LLM's training data. To obtain representative estimates, PoSSUM employs Multilevel Regression and Post-Stratification (MrP) with structured priors to counteract the observable selection biases of social-media platforms. The protocol is validated during the 2024 U.S. Presidential Election, for which five PoSSUM polls were conducted and published on GitHub and X. In the final poll, fielded October 17-26 with a synthetic sample of 1,054 X users, PoSSUM accurately predicted the outcomes in 50 of 51 states and assigned the Republican candidate a win probability of 0.65. Notably, it also exhibited lower state-level bias than most established pollsters. These results demonstrate PoSSUM's potential as a fully automated, unobtrusive alternative to traditional survey methods.