My question very simply is if someone's first name ends in a vowel and their middle name begins with another vowel, would that middle name be properly written or pronounced with a 'h' when said together?

E.g. does Síle Áine become Síle hAíne?

No, h-prothesis only occurs in special circumstances. E.g. in surnames, after Ó (e.g. Ó hUiginn, Ó hAonghusa).

But something else happens:
Síle ends in a schwa-sound /ə/ "uh". This sound is elided whenever another vowel follows - as if written "Síl’ Áine"

/ʃiːl′ə/ + /​ɑːn′ə/ = /​ʃiːl′ ​ɑːn′ə/

You would never say /ʃiːl′ə ɑːn′ə/ (except you speak very slowly with long pauses inbetween)

BTW:
The same happens everywhere,
e.g. Is í Síle atá ag léamh (Síle is reading) /ʃiː ​ʃiːl′ ət​ɑː .../
or Is í Áine atá ag teacht (Áine is coming) /ʃiː ɑːn′ ət​ɑː .../

Go raibh maith agat. That's a great explanation. I was sure something would happen in the speaking of it and wanted to make sure I knew the written form. As it is, there is no alteration to the written form.

So if I were to address that person or refer to them while speaking Irish, Síle Áine might sound effectively like one word, Síláine?

With a slender l. So it would sound like: síleáine.

Of course, if addressing the person, you would have to use the vocative particle (which causes lenition): 'A Shíle Áine'.