I would start with the retailer, that's sort of the "line of command". I started to be in doubt about what I wrote, but eMachines Support, eMachines Support - Answers, E-Mail, Chat - How do I restore my computer using the eRecovery Management program outside of Windows? states, quote:
"Select models, with Microsoft Windows, have the ability to have the Operating System reinstalled from a hidden partition built into the computer."--------
"Note: If your computer does not have an internal optical disc drive, an external optical disc drive will be required in order to recover the Operating System using eMachines Recovery Media.--------
3. Press the CD-ROM drive eject button and insert the disc labeled System CD.
Thus, you either have the hidden partition OR you should have the disk. Please check which one is your case.