How do I get more pet chests in Fallout Shelter?
The Fallout Shelter save games are not particularly robust against editing. It is actually JSON formatted.
You can also just edit in more Pet Carriers using Fallout Shelter Save Editor
The only ways to get the pet carriers are limited.
You get one(as mentioned in the question) from a lunchbox
Missions, it is possible to get lunchboxes, Mr. Handies, and, yes, Pet Carriers
And, of course, buying the pet carries