Is it possible to export the features from a map as geoJSON using an ArcGIS REST service? I'm trying to make a vector layer in OpenLayers based on the output from the map's present view based on a layer.
Not directly, no (see Sean Gillies Blog for some reasons why). To do what you're after the easiest way is to convert from ESRI JSON to geoJSON with some intermediate script/service. See the How to convert ArcGIS Server JSON to GeoJSON? question on this site for a variety of different ways to do this.
One way to publish GIS feature services is to use cartoview www.cartologic.com/cartoview the Cartoview installer will also install GeoNode on windows. After the installation you can download and install the ArcGIS feature server app to serve out Feature services. This way you will be able to serve both standard WMS and geojson as well as feature services from the same platform, which will open the door to work with openlayers or ESRI apps