If the party is on the road, traveling somewhere, and a member is not walking or actively doing anything (ie: being carried or riding a wagon) can they use that time to do downtime activities?
Downtime activities happen between, not during adventures.
Downtime Activities specifically describes time between adventures that characters engage in other non-adventuring activities (DMG 126-131).
Typically characters travel for 8 hours/day (DMG 242-243, compare to forced March PHB 181, and Time PHB 181-183).
Regarding other activities (PHB 183) than navigating, foraging, drawing maps, tracking, and staying alert/hidden and noticing threats during an adventure it is up to the DM's discretion:
However, a character not watching for danger can do one of the following activities instead, or some other activity with the DM’s permission.
So if you want to do other activities such as described in the downtime section, those activities are at the discretion of your DM, but they are not considered downtime activities as these by definition only happen between adventures.