Participatory activities are widely used in human-computer interaction to design with and for stakeholders, but often rely on co-located formats that might exclude geographically difficult-to-reach communities. While remote participation offers a promising alternative to engaging with these communities, we found that existing approaches primarily capture stakeholders' experiences rather than the underlying values that shape them. Especially in sensitive contexts, such as healthcare, eliciting stakeholder values can be critical for informing future technology design. To address this situation, we propose a remote value elicitation approach that combines synchronous and asynchronous activities to foreground stakeholder values, explore value tensions, and examine how values can be enacted. Our work is situated in the context of rare disease (RD), where affected individuals and their families are strained by prolonged, uncertain diagnostic journeys. Based on the conceptual development of our approach with experts in RD care, we critically reflect on broader challenges and opportunities for value-led participatory research in remote settings with difficult-to-reach stakeholders in sensitive contexts.