Praise be to Allah.
There is nothing wrong with ending a letter with the phrase “wa’l-salaam,” and it is not a condition that the phrase should be used in full, because when a written abbreviation is used, what the writer means is the complete expression. So when a person says “wa’l-salaam,” what he or she means is “wa’l-salaamu ‘alaykum.” But if the person sending the letter writes “wa’l-salaamu ‘alayka” or “wa’l-salaamu ‘alaykum” at the end, this is better. Umar ibn al-Khattaab ended his letter to the qaadi Shurayh with the words “wa’l-salaamu ‘alayka” [Sunan al-Nisaa’i, 5304] and ‘Umar ibn ‘Abd al-‘Azeez ended his letter to one of his workers in the same way [Muwatta’ Malik, Kitaab al-jihaad].
Ibn Katheer reported in al-Bidaayah wa’l-Nihaayah from Ibn ‘Asaakir that Ziyaad ibn Abi Sufyaan sent Sa’eed ibn al-‘Aas gifts, money and a letter proposing marriage to his daughter. When the gifts, money and letter arrived, [Sa’eed] read the letter, shared out the gifts among the people sitting with him, then wrote a nice, polite letter back to him in which he said: “In the name of Allaah, Most Gracious, Most Merciful. Allaah says: ‘Nay! Verily man does transgress all bounds (in disbelief and evil deeds, etc.), because he considers himself self-sufficient.’ [al-‘Alaq 96:6-7 – interpretation of the meaning]. Wa’l-salaam.”
However, the person sending the letter should greet the addressee with the complete phrase (“Al-salaamu alaykum”) at the beginning of the letter, as has been the habit of the Muslims from the time of the Prophet SAWS (peace and blessings of Allaah be upon him) until the present. And Allaah knows best.