Is there a service that allows me to draw on an OpenLayers map, and see the resulting GeoJSON?
This will allow you to make live changes and see them on the map in the same window.
Your question is a bit ambiguous. Are you looking for an existing service, or want to create one? Or are you trying to set something up for your users to output GeoJSON?
OpenLayers natively supports GeoJSON and can write to it as well. You should look at this example and see how they write out to different formats. It won't be difficult at all to extend and add GeoJSOn to the dropdown.
You can do this in
QGIS. You need to first download and install it, and then install the
Openlayers plugin. Add a layer from OpenLayers and zoom to your area of interest (it can sometimes take a few seconds to load and update the layer). Then, you can create a new vector layer, click on the editor icon, and draw over the basemap. You can then save your layer as a GeoJson (right click on layer, and choose GeoJson).