Large language models (LLMs) and agentic systems have shown promise for clinical decision support, but existing works largely assume that evidence has already been curated and handed to the model. Real-world clinical workflows instead require agents to actively seek, iteratively plan, and synthesize multimodal evidence from heterogeneous sources. In this paper, we introduce ClinSeekAgent, an automated agentic framework for dynamic multimodal evidence seeking that shifts the paradigm from passive evidence consumption to active evidence acquisition. Given only a clinical query and access to raw data sources, ClinSeekAgent gathers evidence by querying medical knowledge bases, navigating raw EHRs, and invoking medical imaging tools; refines its hypotheses as new information emerges; and integrates the collected evidence into grounded clinical decisions. ClinSeekAgent serves both as an inference-time agent for frontier LLMs and as a training-time pipeline for distilling high-quality agent trajectories into compact open-source models. To validate its inference-time effectiveness, we construct ClinSeek-Bench, which pairs Curated Input reasoning from fixed pre-selected evidence with Automated Evidence-Seeking over raw clinical data. On text-only EHR tasks, ClinSeekAgent improves Claude Opus 4.6 from 60.0 to 63.2 overall F1 and MiniMax M2.5 from 43.1 to 47.3, with positive risk-prediction gains in 7 out of 9 evaluated host models. On multimodal tasks, ClinSeekAgent improves Claude Opus 4.6 from 47.5 to 62.6 (+15.1); all evaluated models improve across the three CXR-related task groups. We further validate ClinSeekAgent as a training pipeline by distilling agentic evidence-seeking trajectories into ClinSeek-35B-A3B, which achieves 34.0 average F1 on existing AgentEHR-Bench, improving over its Qwen3.5-35B-A3B baseline by +11.9 points and approaching Claude Opus 4.6.