I use VimNotes and I often launch Vim and type :Note NoteName, just for editing the same note. I could map a special key for it, but this repeating process got me thinking if I could run a Vim command when bringing up Vim.

Vim likes many other programs, such as Bash or Python, it has -c option. So I can simply do the following:
vim -c 'Note NoteName'
or
vim +Note\ NoteName
That does it. You can supply up to 10 commands. The commands will be executed after (first) file is loaded, if you tell Vim to open a file or files. Also, another --cmd is available but it is executed before vimrc.

If you run Vim in server mode like me, you will need to use --remote-send:
vim --servername SERVERNAME --remote-send ':Note NoteName<CR>'
It is basically emulating your key-presses, so you may need to modify it accordingly. There is also a --remote-expr, you might find it interesting.

I won't use these, at least not for opening my note. I may set up an shell alias, but a key map in Vim is a better option. However, there may have some chances that you really need to run Vim commands in this way.