To do Paypal stored credit card payments, does the app requires to go through any PCI or QSS compliance certification process?
Note: I am using Paypal vault to store credit card info with paypal.
If you are using PayPal Vault then you're good to go. Just make sure you aren't saving any credit card details in your database, in log files, or anywhere else.
If you follow those procedures and for any reason you do have to apply for some sort of PCI compliance (usually not the case) then you'll be able to pass quickly and easily with their low-price method.
You shouldn't need to worry about that at all, though, unless you're doing something with hardware maybe or if you're dealing with a particular company that requires it.