Kate Forbes and Ash Regan made the call amid concerns over the transparency and integrity of the ballot process.
However, Humza Yousaf, the third candidate, who is seen as the establishment choice, warned against indulging in “baseless smears”.
All three candidates have called on the SNP to release its membership numbers, which the party is expected to do later today after initially refusing.
SNP MSP Michelle Thomson – who is running Ms Forbes’ campaign – said some concerns are “based on hearsay or are from bad faith actors”, but she added: “However, others are being openly expressed by individuals within the SNP of longstanding.”
She argued that the fact questions are being asked at all “can only further undermine trust in SNP HQ”.
She added: “We all agree that the party must unify around any newly elected leader. I have asked that the SNP appoint a robust, experienced, third-party auditor of both the ballot processes and the eventual tally of the vote. This third party must have full oversight of all membership numbers, data and processes. This should be done without delay.”
Kirk Torrance, who is advising Ms Regan, said on Twitter: “Independent auditors will give reassurance to the SNP members and the general public that this ballot is conducted properly. This should have been the case from the outset.”
The election process is being carried out by Mi-Voice, an election services firm based in Southampton.
Ms Thomson said the SNP is “ultimately accountable and responsible for many of the processes, oversight and ultimate integrity of the ballot”.
But Deputy First Minister John Swinney, who is backing Mr Yousaf, said the race is being conducted “100 per cent” properly.
Speaking to BBC Radio Scotland, Mr Swinney said: “I don’t understand half of these comments. The process is being conducted by an independent ballot services company, that is an independent organisation. They have got a professional reputation to deliver ballot services effectively and without question.
“My experience, and I have got a lot of experience of elections within the SNP. I have been a candidate in many of them, I ran many of them as the party secretary over many years.
“I have also seen this process we are going through just now operating in the election of previous deputy leaders of the party which we have undertaken in recent years and nobody, nobody, has raised an issue about the authority of those processes.
“They have been completely straightforward, efficient, independently organised processes, and this is exactly the same. So I don’t know what this is all about.”
Asked if he is satisfied the election is being conducted “entirely properly”, Mr Swinney said: “100 per cent.”
Writing on Twitter, Mr Yousaf said: “My team has called for publishing membership figures. I have confidence in the process that uses an external third party company, a process we have used for many years. I am up for challenge and reform of the party, but let's not indulge in baseless smears.”
Voting began earlier this week in the race to replace Nicola Sturgeon as the party’s leader and First Minister, with members having the choice of voting electronically or on paper. However, the exact size of the electorate remains unclear.
The Mail On Sunday newspaper reported at the weekend that 78,000 online forms were being sent out, though the party has not commented on the figure.
Figures from the Electoral Commission show that in December 2021, the SNP had around 104,000 members.
An SNP spokesman previously said: “Candidates have already been made aware that responsibility for the leadership election does not rest with any member of staff.”
The candidates have been told that the number of votes, percentage share, and turnout will be published when the result is declared.