Install El Capitan OS on a new SSD without USB driver
No, you need disk or USB stick to install it.

Why can't you use a USB stick? If that's because you can't buy it from Apple Store, you could use any 8GB+ stick to install drive.

It's very simple, you only need an Install OSX *version*.app file and a USB stick.

Prepare stick


Plug it into your MacBook
Open Disk Utility (Launchpad -> Other)
Choose your stick in left hand side (It should be under External label)
Click on Erase button at top
Choose GUID Partition Map under Scheme, OS X Extended (Journaled) under Format, and name it OSXInstall (You can give it any name, but you'll need to change terminal commands too)
Click on Erase




Get OSX Installer App

For El Capitan open this link, and download it (You may need to click on View in Mac App Store 

You do not need to install it, just download. 

Create bootable installer


Be sure to check that stick is plugged in and shows up in Finder.
If you have no password, set it temporarily 
Open Terminal app (Launchpad -> Other)
Copy and paste this text into Terminal and hit Return: 

sudo /Applications/Install\ OS\ X\ El\ Capitan.app/Contents/Resources/createinstallmedia --volume /Volumes/OSXInstall --applicationpath /Applications/Install\ OS\ X\ El\ Capitan.app --nointeraction

Type you password (It would be hidden, and cursor would not move)
Now just wait till it says Done.


Installation

To boot from the installer stick, hold down the Option key during Mac system start, and select it from the startup volume menu.