Short answer by "rules as written:" yes
Long answer: everything I read in BTB seems to assume wagons must be fielded during most scenarios, from the description of Christian Templin prepping his Merchant Caravan warband (pg 100) to all of the other mentions of wagon (139 of them!). The most relevant parts are these:
pgs 185-186 FAQ about Merchant Caravan, Trade Wagon & Additional Wagons. Note that this section states that a wagon MUST be assigned to a warband member's equipment list, however I find it very curious that the official responder had already made house rules exempting wagons from being assigned to a certain warband member.
pg 48: in the "Horrors of the Underground" scenario, there is a specific exemption for wagons being fielded.
pg 146: the rules about Merchant Caravan warbands including at least one Trade Wagon and what happens if it is lost during a scenario.
pg 182: The Opulent Coach has a specific rule where the warband may leave the Opulent Coach at their camp. This specific rule is not stated for Trade Wagons.
This seems kinda weird, where a traveling warband in northern Cathay would keep their Trade Wagons close during dangerous battles, however would leave an Opulent Coach unattended back at their camp. Still, it seems clear to me that the authors of BTB intended those Trade Wagons to be on the battlefield during scenarios.