There is no in-universe reason for it, in the 2003 anime, the 2009 anime, or the manga. None of them give a reason for it, nor does any Word of God. In fact, Sig Curtis in Brotherhood sparkles a couple times as well, in addition to the rest of the family.
It's worth noting that there is a trope called "Bishie (bishōnen) Sparkle". I doubt many people would classify Armstrong as bishōnen, but that article does mention that it does represent the personality of the one who sparkles (as a bit of a cutesy "he's beautiful inside" fluff). Here's an excerpt from that very article:
- In Fullmetal Alchemist, THE ART OF SPARKLING BEAUTY HAS BEEN PASSED DOWN IN THE ARMSTRONG LINE FOR GENERATIONS! And it really has, although Major Armstrong is more 'manly' than pretty.
- It's not just Major Alex Louis Armstrong who has this trait, but THE ENTIRE ARMSTRONG FAMILY THROUGH THE GENERATIONS.
- Not just the entire family: the sparkle is possessed by a flower woman whose family has SERVED THE ARMSTRONG LINE FOR GENERATIONS. And one of her hairs has the Armstrong curl, too.
- Sig Curtis also develops a sparkle when he's doing manly things with Armstrong. Curiously enough, his sparkle isn't pink, but brown-ish.
- May's Inner-Mind Theatre version of Edward and later Alphonse have this.
- Roy Mustang gets some sparkles in the 13th episode of the 2003 anime adaptation during his duel with Edward.
- As does Envy in the Chibi Wrap Party OVA, to hilarious effect.
- Lampshaded by Breda when he's impersonating Armstrong to prank his colleagues: He's got small pink paper cutouts that look like Armstrong's sparkle, suspended on wires.
Armstrong also sculpted a statue of himself in The Movie from the 2003 anime series; it had a small four-pointed piece of stone for a sparkle suspended by a strip of metal coming from the body.
- Fullmetal Fantasy, the Fan Film produced by the English vocal cast, has Real Life Armstrong complete with sparkles, quite possibly the film's Crowning Moment of Funny.
Also, inarguably, it's funny. And the Rule of Funny applies.