I have a D/S/S/C party, so I have no one who can get the Disarm skill. Currently, one of my NPCs is a Chimney Sweep (the other is a Scholar). Any ideas on how to handle this particular problem? I'm not looking forward to opening chests this game, at the moment...

Telekinesis (eventually).

You don't really have an alternative to getting blown up until you get telekinesis, and even then you're going to get blown up by some chests because there isn't enough space to see the chest but be far enough away to avoid the trap damage with telekinesis (ex. hidden chests in Markham's manor).

Ditch the chimney sweep. He's a waste of money for no real gain. The odds are very small that the sweep's luck bonus will make a noticeable difference in trap damage.

You'd do better getting a healer NPC, if anything. Expensive, but they can get your characters back to full hp for the next chest to explode in your face. Or just heal with your cleric and Druid, and carry lots of food so you can sleep when you run out of mana.

Have an area in you backpack for rings, medallions, belts that increase disarm skill.. Anything that will upgrade the character (for me knight). Switch before opening chests. Sometimes works.
Telekinesis: there does seem to be some sort of physics in the game. So if you move to another room and try to open the chest with your party partially hidden by the doorway, you seem to (sometimes) avoid damage.
Edit: (Probably not necessary. But Protection spells (fire water etc) should be active. )

None of his characters can even learn Disarm Trap.

Sorry I mis-read the original team. Two S's. Light Armor, weak constitutions. Cleric would have to have a high constitution to keep the party from being destroyed. Druid would need to be an equal healer to the Cleric.
So. Use what physics the game allows with telekinesis. Don't try the Haunted House Chests until later in the game because they would be a quick game ender. For those you would need Telekinesis from another room.

Ok, that's all what I was afraid of. Well ... onward!