The Williams sisters have set up a highly anticipated encounter at Indian Wells, where they have never faced off before. Vickery will face Naomi Osaka, who upset Aggie Radwanska after defeating Maria Sharapova in the first round.
In other matches, fifth seed Dominic Thiem beat Greek 19-year-old Stefanos Tsitsipas 6-2, 3-6, 6-3 while South African seventh seed Kevin Anderson overcame Russian Evgeny Donskoy 7-5, 6-4.
Kvitova won the Qatar Open last month and has defeated six top-10 players this year.
The reward is a match-up with older sister Venus, after the eighth seed saw off Sorana Cirstea 6-3 6-4.
There was conjecture that her withdrawal had been a family arrangement and Russia's Elena Dementieva said after her quarter-final loss to Venus that she believed the sisters' father and coach, Richard Williams, decided who would win their matches.
In the men's tournament at Indian Wells, Roger Federer's opening match was put on hold as heavy rain swept through the California desert.
Asked about the semi-final match that never happened, Serena said she has tried to forget.
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Jenner, 20, revealed she thought she and beau Travis Scott , 25, were expecting a baby boy before she found out Stormi's gender. She was also reportedly upset he had been seen going to a strip club in December.
Daria Gavrilova is also through to the third round after a straight-sets victory over Yanina Wickmayer.
"Obviously I wish it was anybody else in the draw, literally anybody, but that's okay". "Just have to go out there and see how I am and do my best".
Federer, 36, had taken a set from Federico Delbonis 6-3 with the second tied at 2-2 when play was suspended.
Serena Williams, of the United States, reacts while playing Zarina Diyas, of Kazakhstan, during the first round of the BNP Paribas Open tennis tournament in Indian Wells, Calif., Thursday, March 8, 2018.
The fourth-seeded Elina Svitolina cruised to a 6-4 6-3 win over Mona Barthel.
Serena Williams has described winning her first WTA Tour match since the birth of her child as an "incredible" feeling. No. 1 seed Simona Halep has won a title at the Indian Wells Tennis Garden in 2015.
Serena dug deep to beat Dutch Kiki Bertens 7-6 (7-5) 7-5.
Seeds Madison Keys (15), Magdalena Rybarikova (18) and Anett Kontaveit (28) all lost on Saturday. However, in the time when play was possible Spaniard Fernando Verdasco used his powerful serve to upset third seed Grigor Dimitrov 7-6 (4), 4-6, 6-3.