Is it possible for multiple people, say a group of students, to collectively work on a story map or to create something with Web AppBuilder?
Yes, multiple people can edit a storymap or WAB. Your AGOL account just has to allow users to edit other users or just share the same username.
It depends on the context of your use.
Providing all of your users have seperate ArcGIS Online logins, each person could use the Collector App to gather data. For example, if you wanted to locate certain places over a geographic area. Each student could pick a district and add points to the map on Collector. If you have the right fields in use, you could have gauges and clocks in Web App builder that would automatically hightlight COUNT numbers.
More on Web App builder can be found here. http://doc.arcgis.com/en/web-appbuilder/