The LORD sets the terms for coming near Him, and they reach Israel as spoken instruction rather than something the people work out. Any Israelite may come, and God fixes what he brings.
The word behind offering means what is brought near, so the animal is how a person closes the distance. Limiting it to herd and flock keeps the gift inside a household's livestock, the wealth it lived on, so nothing hunted could stand in its place. Drawing near runs on God's appointment, and the sacrifice that finally opens the way is one God supplies Himself in Christ.