Grobbelaar picked up the injury at last weekend's Emirates Dubai Sevens, won by the Blitzboks for a fifth consecutive and 11th time overall.
He underwent a scan in Cape Town after the team arrived back from Dubai which revealed that he is unfit to play on Saturday or Sunday.
Tiaan Pretorius and Tristan Lleyds were called into camp on Tuesday and according to Springbok Sevens head coach, Sandile Ngcobo, the squad will be trimmed to the allowed 13 players on Thursday.
"It is really unfortunate for Christie, as he was in very good form in Dubai and every player in the team wants to play in their home tournament,” said Ngcobo.
“As always in rugby, an injury affords another player an opportunity and it could be Tiaan or Tristan. Tiaan is a straight swop for Christie as a forward, but we also added Tristan as he gives us a different option, should we decide to tweak the team for this weekend."
Pretorius made his debut for the Blitzboks last year, while Lleyds have been in the wider training squad for the last couple of months.
"There is a good mood in the camp, and everyone is excited about the opportunity to play in front of family and friends. That stems from the confidence gained from the Dubai experience and performance," Ngcobo said.
"We realise that we need to get back to zero though, we had our 24 hours of glory and now it is back to work again. It was a great start and reward for all the hard work over the last couple of months, but the real challenge now starts and that is to deliver those results on a consistent basis."
Ngcobo said the Cape Town performances of the past showed that success in Dubai does not necessarily mean the same a week later.
"We are back playing with smiles on faces and guys having each other’s backs, and that is great to see. Our playmakers are fit and on form and as such, we will be a tough team to play against this weekend.”
The Blitzboks, top on the standings after round one of eight in the HSBC SVNS, will face USA, Great Britain and Ireland in their Pool A clashes on Saturday.