Vickery will face Naomi Osaka, who upset Aggie Radwanska after defeating Maria Sharapova in the first round.
The eighth-seeded Venus beat Romanian Sorana Cirstea 6-3 6-4 in the preceding match.
Kvitova won the Qatar Open last month and has defeated six top-10 players this year.
Ukrainian fourth seed Elina Svitolina avoided a similar demise, beating German Mona Barthel 6-4 6-3.
Serena Williams and Kiki Bertens have played out several close matches in the past - and today wasn't an exception as the resurgent American needed nearly two hours of play to get past the hard-hitting Dutch, who ultimately fell in a tight 6-7, 5-7 second-round match at the BNP Paribas Open.
The seven-time Grand Slam champion won her opening match on Saturday held at Indian Wells, California, in straight sets, 6-3, 6-4, and now faces the victor between her younger sister Serena Williams, 36, and No. 29 seed Kiki Bertens of the Netherlands in the next round, reports Xinhua news agency.
"She looked like she never lost a step", she said.
The sisters will resume their longtime rivalry on Monday. The two met last time in the 2017 Australian Open final, which Serena won.
Defending champ Roger Federer's opening match was put on hold Saturday when play at the Indian Wells ATP Masters was suspended after heavy rain swept through the California desert.
Despite Serena Williams talent and wins, the 36-year-old tennis Titan has been ridiculed for what many deem inappropriate behavior for a woman.
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 O.K.", Serena Williams said of their latest rematch.
"I'm just so happy to be out here", she said. The tennis superstar beat Zarina Diyas of Kazakhstan during her comeback match on Thursday. The No. 15 seed was mistake-prone throughout the second-round match, committing five unforced errors in the tiebreaker.
Serena Williams has described winning her first WTA Tour match since the birth of her child as an "incredible" feeling.
Serena then took center stage, needing just under two hours to dispatch Bertens, of the Netherlands, 7-6 (7/5), 7-5.