Clapboards are designed to go over some form of sheathing. They are relatively thin and it is possible for wind driven rain to enter the building envelope through them.

Putting any form of exterior siding directly over the wall studs is very unusual and, as far as I know, not ever a sound practice. Sheathing provides a secondary defense against weather and is crucial in preventing racking of the stud walls.

If you added insulation from the outside without any sort of vapor barrier on the inside, you're also very likely to quickly have moisture problems in the wall cavity.

Adding sheathing may create some problems with the exterior trim around doors and windows, but I don't see any way simply nailing clapboards to the studs will work out well.

Lapped siding should be applied over sheathing. Shiplap-type siding (where the backs of the boards form a flat plane) like German siding, drop/novelty siding shouldn't ever be applied directly over sheathing.
A vapor barrier is almost always called for, but the details you choose should be based on best practices for your climate.
Shiplaps are best over a rain-screen/drainage plane system, where an airspace is created with vertical spacers between sheathing and siding. Lap siding (tapered clapboard or weatherboards) have an inherent airspace behind them, and work well right over sheathing.
My 1850 house has internal sheathing and weatherboards directly over the studs outside, so they got the structural benefit of sheathing and the traditional-for-this-region directly-nailed siding. My stud spaces were filled with soft brick/mortar "nogging" but that's another story.
Casey

We applied the insulation with the pink side out, so the vapor barrier would be in the inside. When we pulled off the clapboards, there was vermiculite insulation, which of course all fell out. At which point, we could see the lath and plaster from the interior walls.

The sheathing over the stud will definitely cause an issue with the trim. Would it be an option to cut the plywood, (or whatever we use for sheathing) to fit in between the studs? So they they would be flush with studs, therefore the clapboards would fit as they had before.

If the studs are on standard 16" centers, what you propose to do with the plywood would be an incredibly tedious process and I'm not at all sure what you think the plywood would be attached to. Additionally, sheathing of any any type needs to extend across the studs if it is to provide structural stiffening to prevent racking of the walls. If you're going to add sheathing, whether plywood or another type, there's really no way around it: you're going to have to carefully remove then replace exterior trim.

With no information about your location or climate, it's hard to say to what degree, but if you added regular pink fiberglass roll type insulation with the vapor barrier toward the inside, you're still very likely to experience moisture problems. This type insulation only works as intended when the flanges are correctly stapled to the to the studs on the interior, forming a tight seal. Placing the insulation between the studs with only a friction fit will not allow it to provide much if any insulation value. The plaster and paint will function as a sort of vapor barrier, but there are always many openings such as electrical outlets which will permit moisture to enter the wall cavity.

If the studs are on standard 16" centers, what you propose to do with the plywood would be an incredibly tedious process and I'm not at all sure what you think the plywood would be attached to. Additionally, sheathing of any any type needs to extend across the studs if it is to provide structural stiffening to prevent racking of the walls. If you're going to add sheathing, whether plywood or another type, there's really no way around it: you're going to have to carefully remove then replace exterior trim.

With no information about your location or climate, it's hard to say to what degree, but if you added regular pink fiberglass roll type insulation with the vapor barrier toward the inside, you're still very likely to experience moisture problems. This type insulation only works as intended when the flanges are correctly stapled to the to the studs on the interior, forming a tight seal. Placing the insulation between the studs with only a friction fit will not allow it to provide much if any insulation value. The plaster and paint will function as a sort of vapor barrier, but there are always many openings such as electrical outlets which will permit moisture to enter the wall cavity.

Thank you, everyone for the detailed responses. I was thinking crudely on installing the sheathing inbetween the studs with a small bracket/brace. However, after browsing at Lowe's, I found plywood, fairly thin, and by looking at the clapboards on the rest of the house, we would have about a 1/4" leeway before the siding stuck out further than the rest of the trim. So what we may do is use that as sheathing, since taking off the trim wouldn't work well in our case. I'm sure thicker is better, however, the house was originally built with no sheathing. And most of the house is still sided without it under, so we are going to make due.

I understand about the insulation, fortunately, it's not to late to fix the issue. I have read the the pink side of insulation should always face the outside of the wall. If I do it that way, what would I use as a vapor barrier on the inside facing part? We could probably caulk or spray foam around the outlets. There are actually only two. One light switch and one outlet.

We are zone 5, NE Ohio. Gets pretty cold in winter and hot in the summer.

I should think the tyvek would provide some sort of vapor barrier...but for the inside, plaster and paint already provide a sort of vapor barrier..so go unfaced. Then use foam gaskets for the outlets and switches.

Make certain "plywood, fairly thin" is rated for sheathing or at least exterior glue or your compounding your issues yet again.

Good point on the sheathing glue, I will ask next time I'm there. Columbus guy; gaskets, fiberglass insulation (paper side out, stapled to the studs) sheathing, tyvek, then siding. Does that sound proper?

In modern construction, you do from the inside: vapor barrier (usually sheet plastic stapled to studs), insulation of some sort (loose, foam or batts, then exterior sheathing, then tyvek and finally siding.

In most old houses, there was lath and plaster, a void, then sheathing boards, then perhaps rosin paper, and siding. Later on, the void might have been filled with rock wool or vermiculite, as you found.

With your plaster intact, I'd do unfaced batts as easiest, then do the sheathing, then tyvek, then siding. Paint and plaster and wallpaper layers will work as a makeshift barrier...but use the gaskets in electrical boxes. From what I understand, tyvek keeps drafts out, but also allows moisture to escape from inside.

One other point: caulk around your baseboards and window trim on the inside...this will also help with drafts and moisture movement. My plaster is all original, and I had the walls filled with cellulose fill by the city shortly after moving in...that was 25 years ago, and there has been no moisture problem. From what I can recall now, I think there was rosin paper over my sheathing boards...I don't bother with the outlet gaskets because none of mine are in the exterior walls, they are all in the floor or interior walls.

The post is a little old but, I'll throw in my opinion. It was common to build homes with no sheeting years ago. New homes may use plywood only on corners, to keep the walls from racking. My current home has no sheeting....my previous home had no sheeting. Both have stood 140+ years with no problems. My Iowa house had Walnut siding! I sold Tyvek for 20+ years. It is great but don't use in combo with plastic under your drywall or mold and moisture will be in your future. My vote is....nail siding to the studs. No problems unless adding or changing windows.