Chiclayo is a large enough city that you should be able to find a spot in the city that can help you pack the bike. I would start with some of the bicycle shops. Some may do mostly repairs but there will also be some that get incoming bikes or other access to cardboard. A few may also have a web presence where you can try reaching out in advance. The other spot you can sometimes try in advance is tour groups or companies though I've had more success with bike shops.
The other alternative to Chiclayo would be Trujillo which is an even larger city and in my experience I noticed quite a few more western tourists. There is also a "Casa de Ciclista" known for many years that you can likely find on the internet.
I've done similar on previous trips including more recently Paramaribo Suriname and before that Ushuaia, Argentina. The Ushuaia case was a little more unique because company I was with helped in procuring boxes. However, in Paramaribo, I found a local shop. On that trip, I had started in Georgetown Guyana and had carried along packing materials other than the box (e.g. some foam noodles to wrap around tubes, my own tape,...) and then found a local shop that just conveniently had a set of boxes stashed away.