How do you have vim find or search for a file in the path when opening a split?

As romainl said, :sf is the command that performs this action. There is no vertical split find, although you can do the same thing with the slightly longer command :vert sf.

Typing :split | :find<space>R (meaning "open a split then find..." seems to do the trick. Same thing with :vsplit.

Making it a custom function and calling it with a nnoremap would make it even better.

Edit:

I didn't know about :find and :tabfind so I looked them up in Vim's help and I found :sfind which does exactly one of the things you want. Unfortunately I don't see a :vfind or :vsfind.

Look it up: :help sfind.