Do Pyroelectric gemstones really exist? Can they be found on web market?
Yes, pyroelectric gems exist. No, they don't produce electricity all the time. Pyroelectric gems can generate a certain amount of electric potential when cooled or heated.

To cite Wikipedia 


  The change in temperature modifies the positions of the atoms slightly within the crystal structure, such that the polarization of the material changes. This polarization change gives rise to a voltage across the crystal. If the temperature stays constant at its new value, the pyroelectric voltage gradually disappears due to leakage current (the leakage can be due to electrons moving through the crystal, ions moving through the air, or current leaking through a voltmeter attached across the crystal)


A full list of all piezoelectric minerals can be found at the website of the Mineralogical Society of America

And yes, you can buy them. Some, like Axinite, were used to remove hair and lint from objects.

There's no site like www.buypiezoelectricgems.com... So you should google them one by one and look for sellers. Or go to a gem fair.