In the phrase 'first name' can I omit the word name?

Someone’s initials unambiguously refers to their names, so we can talk about someone’s initials without needing to say that they are initials of their names. In fact, saying so would be redundant: “someone’s initials” already specified that, so “someone’s name’s initials” is tediously repetitive.

No. first name initial is not a customary English phrase, and first initial is not an abbreviation of it.

First initial means "the first of her initials" as you would expect in any other context.

There are occasional contexts where "first" could be read to mean "first name", (eg on some forms) but not any that I can think of in ordinary speeh