I think the expectation is that guests will book direct (online or phone) with the local hotel in order to benefit from the voucher. I have asked that same question several times and the answers are inconsistent between hotels and even within a certain hotel - it seemingly depends on who you talk to in the reservation offices - that has been my experience.
If you saw a rate elsewhere that's lower than the best-available rate on the hotel website, I would call the local hotel direct and ask if they will price-match. If they say yes, then book and ask them to add the voucher request.
If they do not price-match then you have no choice but to tell them you will book elsewhere but will need the voucher. Then you will know for sure.
But please do not think of me as an authority on this! I have asked but I am still unsure!