But, unless there is an API for puTTY, it buys you nothing over using the command line.

Now, the X question -- "Why do you want to do this?"

Both vi and putty are not designed to be used as part of a Unix Tool pipeline (at least not easily). They are standalone tools, and bundling them into your application is not a trivial exercise. It sounds to me like a way to give your application user access to tools that have the potential to cause Major Drechlikeit. Have you talked this over in depth with your system and security admins? Do it today.

Handing a running chainsaw to Joe-Random-User-from-off-the-Web is a recipe for a lot of Grief and Pain. Don't do it.