Knowledge-based Visual Question Answering (KB-VQA) requires models to combine image understanding with external knowledge. Most prior methods use a fixed retrieve-then-generate pipeline with a pre-selected retriever and a static top-k setting, which is not adaptive during reasoning. We propose ProMSA, a progressive multimodal search agent for KB-VQA. Given an image-question pair, the agent iteratively chooses image search, text search, or stop, under explicit tool-call budgets and with deduplication to avoid redundant retrieval. For training, we first use rejection-sampling SFT to learn valid tool-use formats, then optimize the agent with TN-GSPO, a sequence-level RL objective that normalizes updates by both generation length and tool-interaction depth. Experiments on E-VQA and InfoSeek show consistent gains over strong RAG and agent baselines, and improved retrieval and end-to-end accuracy. The code is available at https://github.com/DingWu1021/Promsa.