As far as I understand Emacs doesn't add "~/.emacs.d/elpa" to
`load-path`, when started with -Q flag, but this bug report needs a
fully established `load-path`. So to reproduce the bug please create an
empty ".emacs" file and start Emacs.

Then:

- C-x C-f ~/.emacs
- M-x flymake-mode
- in .emacs buffer type require expression, requiring any library on
  your elpa path, e.g. `(require 'dash)`.

Observe that flymake reports "Cannot open load file: No such file or
directory, dash", even though `load-path` contains
"~/.emacs.d/elpa/dash-<version>/" directory. Screenshot below
illustrates this: