We have a number of different users who want to edit 'sections' of our website. This mostly aligns to Organic groups.
Is there a way to allow people to edit pages of a group which they are members of?
You could, for sure, use the Group module to let users edit something if they are a member of some group. Group allows for creating arbitrary collections of your content and users on your site, and grant access control permissions on those collections. It is available as of D7, and has a D8 version also. It is designed to be an alternative to Organic Groups. The Group module creates groups as entities, making them fully fieldable, extensible and exportable.
For this specific question, you'd enable the
gnode submodule, and for each group type you would define the appropriate permissions (view, edit, delete, etc) for the various Content Types. That's it.
For more details about how Group compares to Organic Groups, refer to my answer to "What are the features of the Group module versus Organic Group module?".