RAW, it's unclear, but most likely nope.

Sneak attack:

Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a Finesse or a ranged weapon.

The Bite from the Ixalan Vampire is a "melee attack", without any specification about being a Finesse weapon attack. The "default" is that weapons do not have the finesse property, unless specified.

The Bloodthirst Feature states

Bloodthirst. You can drain blood and life energy from a willing creature, or one that is grappled by you, incapacitated, or restrained. Make a melee attack against the target. If you hit, you deal 1 piercing damage and 1d6 necrotic damage. The target’s hit point maximum is reduced by an amount equal to the necrotic damage taken, and you regain hit points equal to that amount. The reduction lasts until the target finishes a long rest. The target dies if this effect reduces its hit point maximum to 0.

The emphasized part is the only part that indicates something about the nature of the attack (i.e. it's melee). We are left to wonder about everything else - which led to this question. It's clear that it is not a ranged weapon. As it's similar to a Bite attack, it depends on whether the DM considers the fangs a "finesse weapon" or not. As I said, the default is that they are not, as it is not specified.

Note that the Plane Shift: Ixalan article has the disclaimer

The game mechanics in this supplement are usable in your D&D campaign but are not fully tempered by playtests and design iterations. For these reasons, material in this supplement is not legal in D&D Organized Play events.

similar to Unearthed Arcana (though with different purposes - UA is supposed to be playtested by the other tables). This means the writing is indeed inconsistent and most of the ruling and interpretation ends up to DM fiat.