I can edit the line vertices I create in Collector, but can't figure out how to edit the vertices directly in ArcGIS online.
I can change the length of the line, but not the vertices in ArcGIS online
Yes you can, with your feature layer that supports geometry editing when you click on the feature and click edit then hover your mouse over the line and when it changes from the mouse pointer to the hand click on the line and it will show the line vertices that you can click and drag the vertices to where they need to be placed.
Our ArcGIS online expert found the answer here: How to remove layers data source M-awareness to publish editable featureServer
My line feature class was m-aware, so I had to copy the shapefile into a new geodatabase and disable the m and z values
You can definitely edit vertices in a Web Map. When you publish your map are you making sure that mobile data and feature access are selected? Here is a tutorial I found related to your topic : http://blogs.cornell.edu/onlinegis/2013/07/11/arcgis-online-training-session-2/