Growing interest and investment in the capabilities of foundation models has positioned such systems to impact a wide array of public services. Alongside these opportunities is the risk that these systems reify existing power imbalances and cause disproportionate harm to marginalized communities. Participatory approaches hold promise to instead lend agency and decision-making power to marginalized stakeholders. But existing approaches in participatory AI/ML are typically deeply grounded in context - how do we apply these approaches to foundation models, which are, by design, disconnected from context? Our paper interrogates this question. First, we examine existing attempts at incorporating participation into foundation models. We highlight the tension between participation and scale, demonstrating that it is intractable for impacted communities to meaningfully shape a foundation model that is intended to be universally applicable. In response, we develop a blueprint for participatory foundation models that identifies more local, application-oriented opportunities for meaningful participation. In addition to the "foundation" layer, our framework proposes the "subfloor'' layer, in which stakeholders develop shared technical infrastructure, norms and governance for a grounded domain, and the "surface'' layer, in which affected communities shape the use of a foundation model for a specific downstream task. The intermediate "subfloor'' layer scopes the range of potential harms to consider, and affords communities more concrete avenues for deliberation and intervention. At the same time, it avoids duplicative effort by scaling input across relevant use cases. Through three case studies in clinical care, financial services, and journalism, we illustrate how this multi-layer model can create more meaningful opportunities for participation than solely intervening at the foundation layer.