Q:

How to say "I vaunt a broad knowledge of the subject" without bragging?

How can I say  

"I vaunt a broad knowledge and understanding of the subject"

without sounding too arrogant?
The verb to vaunt implies bragging, in my humble opinion.

A:

Vaunt doesn't imply boasting, but states it outright ("to boast, to brag, to use vainglorious language").
The simple "I have a broad knowledge and understanding of the subject" should suffice. The main thing that will make it not seem boastful is the grounds you have for making the claim.
Ironically, "We are able to boast of a broad knowledge…" can sound less arrogant even though it claims to be boasting, because it suggests a degree of self-awareness. It would though IMO be more boastful in the singular than the plural (because if you are boasting as part of a team, then that can suggest your pride in your colleagues rather than personal arrogance), and as such I'd be more cautious of using it with I than with we.
Perhaps most modest, is to focus on the process:

My experience with the R&D development of MadeUpCo and our work with other industry leaders through MadeUpIndustryBody has given me a broad knowledge and understanding of the subject.

It shows that your claim comes from more than just wishful thinking.

