The contents of the F5 list is encoded in the g:Tex_PromptedEnvironments
variable in vim, adding (changing and expanding the list of environments as appropriate)
let g:Tex_PromptedEnvironments='align,align*,itemize,enumerate'
into .vimrc
changes the list of prompted environments, allowing customisation according to needs. Any environments defined in the tex file itself will also be appended to the list though this may require restarting vim.
If adding an environment which requires something beyond the \begin{foo} \end{foo}
and is not recognised by vim-latex by default, this can be customised with the g:Tex_Env_foo
variable as described in this answer https://tex.stackexchange.com/a/127208/ and in the vim-latex manual http://vim-latex.sourceforge.net/documentation/latex-suite/customizing-macros.html#Tex_Env_name
To set up the itemize environment for example (though this should already be recognised and sensibly processed by vim-latex alongside a number of environments outside those listed by default in g:Tex_PromptedEnvironments
) the following example can be used
let g:Tex_Env_itemize="\\begin{itemize}\<CR>\\item <++>\<CR>\\end{itemize}"