Anna Pinenga said: "Second show for Auckland please?!!! PLEASE!!!??? There are a sad faces on the internet :("
Sammy B Baker echoed that call, asking Chugg Entertainment to put a second Auckland show on the bill for the band's many disappointed fans.
Jared Henry blamed the Ticketmaster website after missing out on tickets.
"I spent an hour refreshing on Tuesday and now 45 mins on here today - nothing, sort out you're (sic) site and stop wasting our time please."
Chugg Entertainment has already warned no further New Zealand or Australia shows will be added to Radiohead's tour.
Its statement announcing the tour dates said: "These are the only Australian & New Zealand shows - no further dates to be announced. Do not miss out."
The promoter is yet to respond to questions about whether another show could be added.
Radiohead have sold more than 30 million albums since their first release Pablo Honey in 1993.
They are known for constant shifts in style, moving from 1990s rock releases The Bends and OK Computer to the stark electronica of Kid A in 2000.
In recent years they have become leading proponents of digital marketing, with fans allowed to pay what they wanted to buy the 2007 album In Rainbows from the band's official website.
The Vector Arena show is the band's first in New Zealand since 1998.
New Zealand-born and London-based musician, Connan Mockasin is playing support during the tour.