I am also having the same problem....
I am using my partner subscription and have over 2400 credits (currently)
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- All the feature classes are shared
- All the services definition files are shared
- the desired Web Map is also shared
Tim suggested to Natasha that she upload a screenshot, but it wouldn't be clear what to screenshot because the "Make a Web Application" option isn't shown....makes it hard to choose a relevant screenshot!
Are you clicking the share button when the webmap is opened? Or are you clicking the share button from the thumbnail/list?
To get the Make Web Application or Embed map buttons to appear you have to hit the share button from within the webmap itself.
It can get confusing since there are about 3 share buttons and only one of them allows you to make an application.