A veterans' legal organization has written VA Secretary Robert Wilkie protesting the stay and asking him to reconsider.
Although a VA official said the department would issue new rules within 90 days, it did not.
The new law will fast-track disability compensation to eligible veterans.
H.R. 299 will extend disability compensation to military personnel who served off the coast of Vietnam.
On Saturday, 536 deceased veterans will be inducted into the Vietnam Veterans Memorial Fund's "In Memory Program."
The unanimous vote paves the way for President Trump's signature.
The move paves the way for more Vietnam veterans to receive disability compensation.
Some veterans organizations take issue with the scope of the legislation.
Lawmakers work to ensure that the VA provides benefits to troops who served off the Vietnam coast.
The Institute of Medicine has linked bladder cancer, hypothyroidism and Parkinson-like symptoms to the herbicide exposure.