I have not been to many in the US, but the best inner park system was in Yosemite California. They had regular trams that complete the valley loop running by stations every 15 minutes.

Not a national park per se but you can take a bus from downtown Portland, OR to the Columbia River Gorge / Multnomah Falls and Mount Hood!

That along with plentiful hikes in Washington Park makes the area quite accessible for hiking and outdoor activities without a car.

There’s a guy on YouTube who has a series where he travels to different national parks (and other places) via mass transit. It’s pretty fascinating, generally.