The answer is "yes."
I want to install a new Solid State Drive (SSD) into the Mac. Can I just build in the SSD, start the MacBook in Recovery Mode and install OS X from there on?
It's called Internet Recovery. You basically hold ⌘ Command⌥ Option R to launch an installer sourced from Apple over the Internet.
OS X Daily has a good write-up about it.
I have done this personally on several MBPs. One caveat to keep in mind is that it will install the latest version of OS X that is linked to the account/machine. If it was Mavericks or Yosemite, that's what you will get. If you didn't have El Capitan, you will have to upgrade to it once the install is finished.
Internet Recovery is also a bit slow (dependent on your Internet speed/bandwidth). What I have done is create a USB installer of my image and then put it away for safe keeping. When I have to fix someone's Mac and reinstall, instead of going though the hassle of installing an out-of-date OS X then upgrading to the latest, I just install the latest from scratch via USB.
Create a USB Installer
After downloading the Installer Application, Using a blank USB flash drive (8GB or larger for El Capitan, 16GB or larger for Sierra/High Sierra), issue the following command (replace "Untitled" with the name of the volume that gets mounted if different):
sudo /Applications/Install\ macOS\ Sierra.app/Contents/Resources/createinstallmedia --volume /Volumes/Untitled --applicationpath /Applications/Install\ macOS\ Sierra.app
The example above uses Sierra as the Installer. The instructions are the same for El Capitan and High Sierra with the only exception being the path; modify accordingly.