Could a person suffering under a vampire's charm have said charm dispelled?
Or that of a succubus?
Yes and yes. Both are magical effects as I read it and can thus be dispelled.
Though other DMs may rule differently.
I was thinking that too, but then I read the Dispel Magic description and it specifically mentions "spells" not "magical effects." :/
The first line of the spell reads: "Choose one creature, object, or magical effect within range."
For sure. The next two sentences though "Any spell of 3rd level or lower on the target ends.
For each spell of 4th level or higher on the target, make
an ability check using your spellcasting ability. The
DC equals 10 + the spell’s level. On a successful check,
the spell ends."
...which is why there's confusion about it in my group.
The charm ability of the succubus or vampire a magical effect. Dispel magic targets magical effects. If the DM feels that the use of dispel magic will have an uncertain outcome, he or she can assign a DC and call for an ability check.
Choose one creature, object, or magical effect within range. Any spell of 3rd level or lower on the target ends. For each spell of 4th level or higher on the target. make an ability check using your spellcasting ability.
No. Dispel Magic targets any spell on a magical effect.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.