Is there an equivalent to emacsclient for vim?

I actually figured this one out on my own. This is supported in vim natively if you use the --remote option. Ex:

gvim --remote foo.txt

You can also specify a server name:

gvim --remote foo.txt --servername foo

I added the following function to my .zshrc (but it should work in bash as well) to make this a bit easier:

function vs() {
    gvim --remote-silent $@
}

This way, all you have to type in is vs <filename>.

Is there an equivalent to emacsclient for vim?

Not really.

I want to have something I can type at the command prompt to open a file in a vim window I already have open rather than creating a new one

This can be done with gvim --remote.

But emacs client is much more useful than just gvim --remote. With emacs client you can open a duplicate emacs gtk window. Then you can have emacs open with some files on your machine, ssh in with X11 forwarding and run emacsclient to pop up a emacs window on your remote machine. It's like screen for emacs. Unfortunately, gvim --remote only works to evaluate expressions, send keybindings, and open files, which while useful and matching your usecase, is not the same as emacsclient.