When user queries are incomplete, large language models should ask targeted questions before committing to an answer, a capability we call proactive elicitation, yet evaluation on real user-assistant queries across seven LLMs (8B to frontier) shows that fewer than 11% of underspecified queries trigger a follow-up question. To benchmark and train this capability, we introduce VagueBench, where each seed query is annotated with structured intent and detail units then degraded into intent-vague and fully-vague variants via controlled removal of linguistic cues, yielding known hidden information sets for evaluating both when and what to ask. We propose DialogGRPO, an offline reinforcement learning method that organizes multi-turn elicitation as a dialogue-state directed acyclic graph, where candidate questions at each node receive dense turn-level rewards combining immediate information gain with lookahead value through group-relative comparison, enabling fine-grained credit assignment without runtime simulation. DialogGRPO elicits missing information on over 88% of vague queries with substantial gains in both recovery efficiency and answer coverage, without over-asking on complete queries, outperforms prompting and test-time-scaling baselines, and generalizes zero-shot to MultiWOZ 2.4 and AmbigQA.