All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
There is saying which is widespread among the common people who attribute it a Hadeeth whose wording is “There is no Salaam (i.e. no greeting) whilst eating.” However, this is not a Hadeeth as stated by the scholars . Al-Ajlooni wrote in his book "Kashf Al-Khafaa’": “There is no Salaam whilst eating.” is not a Hadeeth.”
In principle, it is legislated to initiate Salaam and to respond to the one who gave Salaam. Initiating the Salaam is a Sunnah (recommended) and responding to the one who gave Salaam is an obligation, this is whether or not a person is eating for the general evidence in this regard, like the saying of Allaah (which means): {And when you are greeted with a greeting, greet [in return] with one better than it or [at least] return it [in a like manner]. Indeed Allaah is ever, over all things, an Accountant.} [Quran 4:86]
Abdulllaah Ibn 'Amr, may Allah be pleased with them both, narrated that a man asked the Prophet ''What in Islam is the best?” He [the Prophet] said: “To feed people and to say Salaam to those whom you know and those whom you do not know.” [Al-Bukhari and Muslim]
In principle, one should stick to the general evidence unless there is evidence that specifies it.
However, some scholars, like An-Nawawi said that the previous statement has a correct meaning if what is meant is shaking hands or responding to the greeting when a person has food in his mouth.
Allaah Knows best.