Once you have paid you will then receive an email from PIA. There is no download until you actually do a download. In the email will be a user name that looks like "Pxxxxxxx" i.e. P1234567. If you did not receive an email from PIA then check you junk folder or spam folder. You ISP may have decided that the email was not legit and they killed it.
If you cannot find the email with login info then contact support and submit a ticket. Putting a post in the forum might not be seen by the Support Personnel.
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If you cannot find the email with login info then contact support and submit a ticket. Putting a post in the forum might not be seen by the Support Personnel.
To answer your question, not, PIA is not a scam.