Is it possible to swap the items you're holding?

I would look at your controls menu to see if there is a specific button because I haven't found anything that labels what exactly could swap two items, but there was this info from Polygon.

In the original Mario Kart 8, holding an item behind your kart wouldn’t remove it from your currently held arsenal. That has not changed in Mario Kart 8 Deluxe, but there are now two active item slots, so you have more wiggle room and can make more liberal use of your items regardless of your current standing.

One of the oldest Mario Kart tricks is to hold an item like a shell or banana behind you to block shots or automatically hit someone who’s tailgating you. Upon impact, the item will be consumed — its job is done, and you can grab a new item. Get in the habit of cycling through objects or putting them in situations where you can use them instead of hoarding them.

The parts I bolded suggests you can swap between the two active item slots, but no indication as to what button(s) perform that action.

However; as GarrettJ pointed out (which is much more likely the case), cycling through them most likely means consuming the items to get through them.

HOWEVER, this thread, suggests you cannot, unlike in the game double dash - whatever is your first held item seems to go into your back (2nd) slot and stays until the 1st is gone.