Mostly no, but some offer free trials.

This requires a download, but it is the only completely free, official version of the actual BeamNG engine available.

Wait for a Steam Sale. BeamNG.drive frequently drops to $15-$20. For the 200+ hours of sandbox play, it is cheaper than a pizza.

You cannot get "free play" here, but you can achieve "no download" if you already own the game.

Occasionally, you might stumble upon a website that actually loads a 3D scene. You might see a car and a road.