It's been ages since I tried a dash cam app. They do exist, but I think most are paid if you want the full features. The one I used was decently featured for the free version, but that could've changed by now.
As for storage, I doubt it'll be a big problem. Dash cams record a rolling set of time. So it will temporarily store, for example, a minute worth of video that constantly updates as it runs. A new second is recorded, an old second discarded. When something happens, you either press a save button or a sudden g-force shock (depending on the app) will save that minute or so video. So in reality, it's using very little space and it wouldn't be a concern unless you have hours of saved video. If you are saving that much video, then I'm guessing you are going to want to offload it to a computer or YouTube, which would keep the phone storage relatively free anyway.