vim shell key mapping to execute shell commands
The vimshell plugin provides an interactive shell inside a Vim buffer. Apparently, you don't need the interactivity (because you intend to immediately exit after issuing the shell command). For that, you don't need the plugin itself; the built-in :! command already allows you to launch external commands:
:nnoremap <leader>bi :!bundle install<CR>If you want to keep the output visible, you can read it into a scratch buffer:
:nnoremap <leader>bi :new<Bar>0r!bundle install<CR>
Having an interactive shell in Vim is one of Vim's stated non-goals (cp. :help design-not), so the plugin has to jump through several hoops to make this possible. Those hacks are causing these problems (of defining a proper mapping, as evidenced by the attempts in the question's comments); lack of automation (like through mappings) is a limitation of this approach.
You may contact vimshell's author (via email or GitHub issue); he's usually very open and responsive! He's in the best position to make such mapping work.