Our 80-year-old cottage has shiplap siding that leaks through the horizontal seams during a rainstorm. To avoid replacing the old shiplap (no rot) can I fill the seams with caulk? Foam insulation? The inside walls are open to the inside of the shiplap, so I can caulk/foam from both inside and outside.

Any suggestions?
Breezy

Sure. Use clear 100% silicone. I'd apply it from the outside, of course trying to get it "up in there" so it's less visible.

Foam is great, but I avoid using it in areas that can be seen.

Make sure all the joints are tight then seal like rickj said but use latex caulking w/ silicone additive. You can't paint 100% silicone.