Very interesting question, and it touches some core "low-level" typography principles.
So here is an informal insight from what I've learned from own experience.
I would name two important factors here.
- Fonts which are exploiting variable stroke width and serifs.
So, here it goes directly into one mysteriuos phenomenon which makes text more readable, namely variable stroke width. For example let's take Times New Roman:
As seen the initaial glyph structure has thicker strokes in the direction shown with an arrow (ca. north-west to south-east).
Now you would probably ask: how this direction is determined?
The things is, this is non-answerable formally, and this direction is determined experementally, and the mystery is, that namely this direction makes better optical results than others, so this is
quite unexplainable phenomenon.
Serifs are also subject to this 'sight orientation dependency', since they are part of the glyphs and similar principles apply.
So here are two possible way-outs:
a) Simple way: use a sans-serif font with constant stroke width.
b) Harder way: develop or find a font which is initially developed with that in mind, namely where strokes width, (for horizontal direction), are thickier in perpendicular direction (north-west to south-east) so when you put it vertically, they will be thicker again in correct direction.
And respectively the serifs which are made specially for vertical text.
Again, quite unexplainable thing, I have noticed that same glyphs, when put vertically can be read easier when one squishes them, i.e. a narrow font variation will be better readable in vertical orientation.
Here is one related optical illusion:
Look at the picture and guess, what is a perfect circle and what is an ellipse here? One probably would say that on the left is a perfect circle, and on the right is an ellipse.
The answer is, both are ellipses, and those are same objects, rotated by 90 degrees.
In other words, if one watches a movie which is stretched horizontally by, say 105% and then watches the same movie stretched vertically by same 105%, then one more probably notices this defect in the second case.
This means, a highly proportional font, like e.g. Perpetua, will be bad choice for vertical orientation, but if made narrower, will work better.
So, as a very simple solution, I would take a narrow sans-serif font with constant stroke width.
To solve this the hard way, one should develop a font which takes the orientation in account, or try to find such a font (I never heard of such, would be exciting to find one).