Do you need a third-party accessibility audit?
This question usually arrives attached to a quote for several thousand dollars and a deadline that does not accommodate it. The honest answer is that it depends on what was actually asked for, and that it is often no.
A self-evaluation is the normal case
Most conformance reports in circulation are written by the vendor about their own product. That is what the template is for, and a buyer asking for a VPAT is, in the ordinary case, asking for exactly that.
What makes a self-evaluation credible is not independence. It is that the evaluation actually happened, that the methods are stated, and that the gaps are in the document rather than discovered later by the buyer.
When a buyer really is asking for more
Some buyers do require an independent audit, and when they do they say so in those words - "independent", "third-party", "by an accredited evaluator". Read the request rather than inferring. If it is ambiguous, ask; the answer takes a day and changes the cost by an order of magnitude.
A third-party audit is also worth buying on its own merits when you have reason to believe your product has significant gaps and you want them found by someone who is not you. That is a different decision from what the buyer is asking for, and worth making separately.
What an audit does not do
An audit does not make you compliant and does not certify you. Nobody can certify a product against WCAG. An audit produces findings, which you then have to write up and act on.
If a vendor offers to make you compliant, or to guarantee you against complaints, that is a claim nobody can support, and it is the same claim that makes a procurement reviewer discount a conformance report.