If his offering is a sacrifice of peace offering, if he offers an animal from the herd, male or female, he shall offer it without blemish before YHWH.
Leviticus 3:1

A peace offering can be any unblemished animal from the bovine, caprine, or ovine categories. Since it is intended to be a shared feast, none of the sacrificial birds would be appropriate.

There is no special provision for the poor. If you couldn't afford a peace offering, you just didn't bring one.