Perturbative treatment of infrared divergencies in QED of arbitrary order

In the comments, OP clarifies that the issue is with equation (13.2.3). I was confused about this part as well when I first learnt it. The proper way to perform the calculation is as follows.

1. First, do the $q^0$ integral by contour integration. We assume that the numerator of the integrand (which involves the lower point amplitude, the numerator of propagators, vertex factors, etc.) does not contain any poles in $q^0$. Thus, all the poles arise from the zeros of the denominator so there are four poles in $q^0$.

2. Note that some poles are $q^0 = {cal O}(|vec{q}|)$ and some poles are $q^0 = {cal O}(1)$ in the small $|vec{q}|$ limit.

3. Show that the ${cal O}(1)$ poles do not contribute to the infrared divergence. More precisely, the ${cal O}(1)$ poles give an integral which is ${cal O}(|vec{q}|^{-1})$. The integral over $vec{q}$ then gives a term that is ${cal O}(Lambda^2)$ which vanishes in the limit $Lambda to 0$.

4. With the foresight that only poles which are ${cal O}(|vec{q}|)$ contribute to the infrared divergence, we can expand the entire integrand in small $q^mu$ (not just small $vec{q}$) and therefore arrive at eq. (13.2.3).