It is permissible to give salaam whilst eating, and there is nothing in the Sunnah to indicate that it is not allowed. The wide spread saying that there is no salaam whilst eating (“la salaam ‘ala ta‘aam”) has no basis in Islam.
Al-‘Ajlooni (may Allah have mercy on him) said in Kashf al-Khafa’: [The phrase] “there is no salaam whilst eating” is not a hadeeth. End quote.
Some of the scholars stated that the expression referred to above has a sound meaning, if what is meant is shaking hands or responding to the greeting when a person has food in his mouth.
Al-Nawawi (may Allah have mercy on him) said in al-Adhkaar: An example of that is if a person is eating and has food in his mouth. If someone greets him in this situation, he does not deserve an answer. But if it is whilst he is eating but he does not have food in his mouth, then there is nothing wrong with someone giving salaam, and he has to respond. End quote.
Shaykh ‘Abd al-Rahmaan al-Suhaym (may Allah preserve him) said: This expression is a saying among the people and it is not a hadeeth. Its meaning is correct if what is meant is shaking hands, but with regard to simply giving salaam, it is not disallowed when one is eating. End quote.
For more information please see question no. 142516.
And Allah knows best.