They can, with a catch and a few assumptions.

The only way RAW I could find to resurrect the victims using the Resurrection spell is by first casting Greater Restoration in order to remove the effect of the vampire's bite (since the victims are not vampires yet), and then cast Resurrection. The Resurrection spell on its own would fail, since the bite is still in effect. This would need to be done within 10 days, as otherwise the effects of Gentle Repose would go away and the victims would rise as free-willed vampire spawn.

This however, assumes that the vampire's bite, or at least it's health reduction component is a magical curse, that Greater Restoration can be cast on a corpse that is soon to rise as an undead without intervention (on the same principles that sleeping creatures are nevertheless creatures, even though the victims in the case would be temporary corpses if left unattended), and that Greater Restoration covers vampirism (assumption made from lycanthropy being able to be cured by Greater Restoration).

Can you cast Gentle Repose multiple times in order to extend the "revivability" of a body as much as you like?

Let's take a look at the text of the gentle repose spell:

Casting Time: 1 action
Range: Touch
Components: V, S, M (a pinch of salt and one copper piece placed on each of the corpse's eyes, which must remain there for the duration)
Duration: 10 days

You touch a corpse or other remains. For the duration, the target is protected from decay and can't become undead.
The spell also effectively extends the time limit on raising the target from the dead, since days spent under the influence of this spell don't count against the time limit of spells such as raise dead.

p. 245 PHB

The answer is yes.

As long as you are able to cast the spell again before the duration ends, you are effectively stopping the clock for the 10-day window of the reincarnate spell. Also, take note there is a stipulation in the spell components that the two copper pieces must remain on the corpse's eyes for the duration.

Does this make a character unkillable?

The gentle repose spell does not have an effect on living characters, so this would, in fact, do nothing to make a character "unkillable."

Keep in mind that casting the reincarnate spell on a protected corpse does not guarantee they will return to life:

Provided that the creature has been dead no longer than 10 days, the spell forms a new adult body for it and then calls the soul to enter that body. If the target's soul isn't free or willing to do so, the spell fails.

p. 271 PHB (emphasis mine)

A creature's soul could be prevented from returning to life for a number of reasons in the time between its death and attempted resurrection.