If your song is really simple, e.g. only consists of the 3 notes that make up a major triad (root, third, fifth), then this is definitely possible and you can just use natural thirds and natural fifths.
But as you start adding more notes, more chords and perhaps change of keys etc, it starts to break down.
That's the reason why J. S. Bach wrote The Well-Tempered Clavier. It's a collection of 24 preludes and fugues, in each possible major and minor key.
The basic idea was that if every prelude and fugue sounded good on an instrument (organ, harpsichord etc.), than it meant that the instrument was "well-tempered".
Using natural tuning instead of 12-TET would have resulted in some pieces sounding very good and other sounding very bad.
You can. It’s called adaptive tuning, or dynamic just intonation, and it happens naturally for singers with no accompanying instruments.
It’s impractical on a real instrument, but there’s a commercial synthesiser implementation called hermode tuning.
You’re trading one problem for another, though. No matter how you do this, you will either have occasional mis-tuning or else your notes will drift.
I used to play fretless bass in a garage hip hop troupe that played with heavily manipulated samples that were all over the place in terms of tuning instead of locked to A440, forcing adaptations like "this section is a minor chord a little above C#".
Adaptive tuning is hard to do on a guitar because the frets are fixed. String bending doesn't help much because the biggest issue is that major thirds are too wide in equal temperament and string bending the third makes pitch go up and exacerbates the problem.
You can do a teeny little bit using lateral pressure (along the string) to move something flat. It's very difficult to make adaptations in chords though. A studio musician trick is to retune the guitar slightly for certain sections, though this can screw with everybody else in the ensemble.
Attempts to experiment with temperament using squiggly frets make it clear how challenging this problem is: https://stringjoy.com/true-temperament-frets-explained/
But with the way I played, I'm not even sure how much it mattered. The best tool for enhancing my playing would've been a mute. (And it would have been most effective lodged in my windpipe.)
Additionally, some songs even change keys, which makes “per-song” not enough of a constraint.
(Though something that happens in just intonation is that you often find out you need more notes than you might have originally thought, because JI makes distinctions between notes that are treated as the same in 12-TET. For instance, you might have 10/9 or 9/8 as your major second, or your minor seventh might be 9/5, 16/9, 7/4, or 12/7 depending on context.)
I don't think any just intonation guitar has been mass produced, but you can definitely build one or modify an existing guitar if you have the right tools and are willing to do a bunch of math and learn how to install frets.
This page is about a JI keyboard I built a while back, but there's also a few pictures of a couple old Harmony guitars I adapted to JI: https://jsnow.bootlegether.net/jik/keyboard.html
Here's a so-so performance of myself playing a Bach piece on a newer and vastly improved version of that just intonation keyboard: https://www.youtube.com/watch?v=rqbWnDhip0A
In 12-EDO the song has 11 distinct pitch classes. (Bach used the tritone, but not the minor second.) In my straightforward JI interpretation, I use 15 pitch classes. (I would have used 16, but my keyboard simply doesn't have a key for that note.)
You can listen to variations here: https://youtu.be/kRui9apjWAY?t=622
0: https://www.guyguitars.com/truetemperament/eng/tt_techdetail...
With relative pitch music sounds the same even if you deviate from the original equal temperament pitch of the key you started singing even changing the key.
For the same reason if there is a fixed instrument playing at the same time, like a piano accompaniment, it's sound would be used as a reference and the singers would not drift
But for instruments with fixed pitches, like guitar or pianos,12 equal temperament is the best compromise to be able to play in all keys.
You might play a G# note in the context of an E chord (where it's the third), and then you might play it in the context of a C# (where it's the fifth).
These are discernably different pitches, but the same "note", in the same key, in the same song!