Q: Some schools keep the back completely erect while others lean a bit forward in the Tai Chi postures. Is there any old documentation as to how straight and upright the body should be in Tai Chi?
A: Yes, the classics in the Li Family manual (and also the other known versions) specifically states that we should “hollow the chest and extend the back”, and sink low. 
Furthermore, all traditional masters of the old styles lean a bit forward in nearly all postures. The oldest pictures we have are of Yang Chengfu, but all other early pictures show the same inclination of the body. The longer steps, the more the body lean.
There is a famous comment on posture in the original classic from 1590:
POEM ON THE PRACTICE OF THE THIRTEEN MOVEMENTS
The set of the thirteen movements should not be taken lightly
The fountain from where its purpose well, is in the waist.
Carefully pay attention to the shifts and turns of empty and full,
Qi must be in the entire body without the slightest deficiency.
In stillness on your own moving like you were still,
but the opponent shifts and vary so this is a marvel.
Each and every move has design and purpose,
but it will not come without consciously spending time and effort.
In every moment keep your mind on the waist,
Be relaxed and quiet in the abdomen, then the qi will rise.
Hold the lower regions centred and straight, then the spirit passes
through to the top of the head.
The whole body light and sharp, HOLDING THE HEAD AS SUSPENDED FROM ABOVE.
Give full awareness to that which you want to learn.
Bend, extend, open and close should be natural.
Go through the gate and seek guidance, the teaching must be word of
mouth. Exert yourself ceaselessly and study the method on your own.
If you ask what the principle for the body is,
the intention and qi is the ruler, bones and flesh are the subjects.
If wondering about the final purpose, it is to prolong life and stay youthful.
This is my poem of one hundred and forty characters,
word by word clear and distinct, nothing omitted.
If proceeding without heeding these words you will waste your effort and
have cause for sighing.
The second spring moon of the Geng Yin year (1590) in the reign of
Great Ming Dynasty Emperor Wan Li. Instructional poem on the thirteen
boxing movements. First time presented in the Great Hall of the
Ancestral Temple. Li Chunmao (Li Yezhen)
Translation Lars Bo Christensen