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.