I often see donation requests with amounts in multiples of $18. I would like to know the source for giving charity of the numeric value of the Hebrew word chai (18) and its multiples.
When charity is given as a form of merit, as opposed to a general tithing obligation, it is seen as an intercessor before God. This is termed a kataigor. Angels also play this role, especially the angel Michael. To attest to this symbolically or perhaps to request that this angel intercede and serve as a kataigor on our behalf there is a custom to give charity with the numerical equivalent of Michael - 101. This custom is still prevalent among Sefardic Jews.
Since giving 101 of a particular currency may not be feasible other numeric equivalents which were connected to angels were found. 18 stems from a kabbalistic understanding of a story in Chagigah 15b: Rebbi Mayer found a pomegranate (רמון) he ate the inside and cast away the husk. The kabbalists understood this to be a veiled reference to the angel מטטרון [MaTat]. The idea being that the letters of the word רמון are the outter letters of the name מטטרון. When those are removed you are left with ט-ט, the inner substance which is what Rebbi Mayer "consumed". The gematria value of ט-ט is 18, hence the giving of charity in multiples of 18.
While not directly answering this question, there is an incident recorded in R. Hayyim Joseph David Azulai's diary that shows that the gematria of the sum of charity was taken to be significant. Once you have that premise, it is not a huge leap to specifically the value of chai, as I think most people would readily acknowledge "life" to be the the most fundamental blessing. Multiples of chai is surely just a way of amplifying the message of "life", (as opposed to an entirely separate rationale). Here is the text of the diary entry for November 21st, 1753:
'And the reply came back' from the eldest among them that they had reached agreement, 'and a spirit of goodwill supported them' for the sum of two hundred and seventy five Corinthian ducats [= escudos]. I answered that 'I went through the sum' mentioned, and 'there is a hint of the number' of evil, and it is not fitting to repay evil for so much good – 'for they had rendered me only good'. And when they heard me present this pleasantry it found favour in their eyes, and they added another five! So I left there with 'behold! it is good.