The PCS, which represents staff working on the system, wants the NAO investigation to look at whether curtailing testing contributed to delays and backlogs when the system was introduced earlier this year.
The spending watchdog should also question EDS about why it took so long to correct faults in the software and whether sufficient resources were devoted to the project to meet its politically-driven deadlines, PCS said.
"The problems that beset the introduction of [the tax credit system] and the pressures put on staff and customers through computer failure, poor planning and allocation of resources seriously undermined what is a flagship government programme," said a union official.
The Inland Revenue dismissed PCSclaims. "When the system was working more slowly than it should have been we were working exceedingly hard with EDS and doing everything possible to stabilise the system and to get it functioning properly," a representative said.
In July paymaster-general Dawn Primarolo admitted the tax credit computer systems had contributed to late payment for hundreds of thousands of claimants.
EDS was not available for comment.