Did I do something wrong during the pregnancy to cause my child to have a brachial plexus injury?

No. A brachial plexus injury is usually caused during a difficult delivery it is not something that happened during pregnancy.
Erb's palsy. Brachial plexus injury is rare, but when it happens it is usually seen when the newborn is very large and the vaginal delivery is difficult. Significant traction on the arm can lead to neuropraxia or axonal injury to the nerves that constitute the brachial plexus, the most common manifestation of which is erb's palsy = the arm is turned inward with the palm facing backward.
No. No this is not your fault. The birthing process if childs shoulder had difficulty coming out of mothers body can cause this issue and you might not have any control over this.