Praise be to Allah.
A sheep whose fatty tail has been cut off is not fit to be offered as a hadiy, udhiyah or ‘aqeeqah, because of the report narrated by Ameer al-Mu’mineen ‘Ali (may Allah be pleased with him) who said: “The Messenger of Allah (peace and blessings of Allah be upon him) commanded us to check the eyes and the ears, and not to sacrifice any sheep that was one-eyed or whose ears had been cut and some of it left hanging, or any whose ears had been cut from the back, or any whose ears had been torn or split.”
Narrated by Ahmad, 2/210, no. 851, and by the four. Classed as saheeh by al-Tirmidhi and Ibn Hibbaan.
This applies if it has been cut, but if the lamb was born without a tail in the first place, then it comes under the ruling of one that is created without horns and one that has a small tail. The ruling is that these animals are good enough for sacrifice.
And Allah is the Source of Strength.