"The family remain hopeful the inquest and investigating will provide the answers they seek," the MP said in a statement.
Mr Vaz said the hospital had told him its own internal inquiry was expected to be completed after the inquest which reopens at Westminster Coroner's Court on Tuesday.
The Labour MP has also written to Austereo in Australia which he noted remained "subject to an investigation by the Australian Communications and Media Authority".
Ms Saldanha's family will be represented at the inquest by John Cooper QC.
He's been acting pro bono and earlier this month hit out at a decision to deny the bereaved family legal aid for the inquest.
"The family believe there are questions to be asked about the hospital protocols that led to Mrs Saldanha being put in that job on that night," Mr Cooper told the Independent newspaper.
Prosecutors in early February announced they would not bring charges against Greig and Christian over the hoax call because there was "no evidence" to support a charge of manslaughter.
- AAP