We have multiple Adobe Cloud team members that need to update a single TypeKit. Can the kit be shared between team members, if so, how?
Each Typekit account licenses the account owner to sync fonts through their personal Adobe ID. It isn't possible to share a single Typekit account with the rest of your office.
Typekit does not have kit sharing capabilities at this time, though we're looking into it as a possibility for the future.
Everyone in your team who has a Cloud account should have equal access to TypeKit. However, the fonts have to be manually installed to each individual computer. So, when one of your users opens a document and gets a message that the font is missing, they can use the "Sync" option in their "Find Font" window, and their computer will go out and search for the missing font in the TypeKit catalog. When the font is found, it will be downloaded to that individual computer, and you won't have any more missing font issues.
By the way, the vast majority of my work is done in InDesign, so the Find Font window that I describe is the one in that program. The other Adobe programs will probably have slightly different functionality. But as long as you can find the option for syncing a font, the process will be essentially the same.