It's apparently not, since it gives the warning at the start. I'm not in incognito mode, and my browser settings should be set to allow usage of localStorage (although that has been acting up a little bit due to some games with horribly unoptimized save files).
I put it down, talked to Fargleton and moved to the next section where I'm taught to move the van around.
EDIT: After manually clearing a slew of data from the hwcdn domain, it works now (it's a fatal flaw of itch's web game system >.<)