The Tamil IME is indeed available from this link. I have installed
it now very easily.

This is what I was looking for. I felt from the beginning that
Tamil must have an IME because of its script characteristics but
thought that it didn't. I thought Microsoft had not offered IME's
for Indic languages, however, I was wrong, they do now. The IME
offers simply a transliteration program, a phonetic Tamil keyboard
designed for the Tamil language, and the inscript keyboard. Now I
know what is what.

Suzanne McCarthy

All I ever wanted to say from the first was that Indic scripts share
a certain characteristic with East Asian scripts that means they can
be appropriately input with an IME.