More than a dozen senators missed the vote.
Sens. Ben Cardin and Bob Corker are hoping to move forward with the legislation.
The Senate Democratic leader said the law had turned out to be a "smashing success."
New amendments set up hurdles as lawmakers work on a path to passage.
The senators said the VA has the authority to grant benefits to C-123 veterans.
Senators are expected to debate Iran legislation Monday afternoon.
Both Corker and Cardin made their opening salvos in support of the legislation.
A Cruz spokeswoman suggested a procedural vote was "the only vote that mattered."
The vote came more than 160 days after she was nominated by President Obama.
But Majority Leader Mitch McConnell (R-Ky.) said there will be debate only this week.