Sure. Go ahead and make an attack roll.
There is no rule that you cannot attack yourself, but all the usual rules for making an attack apply. You must make an attack roll against your own armor class. On a hit, you deal 1d4+CON piercing damage to yourself, and can choose to gain a obnus on your next ability check.
For a more thorough discussion of attacking yourself, see here: Can a PC attack themselves with an unarmed strike?
This works because Wizards removed the feature that prevented it from the UA version.
In the Unearthed Arcana version of the Dhampir, we see:
Type: Humanoid and Undead
Your type was humanoid and undead, meaning the UA Dhampir is not an eligible target for its own bite. Thanks Wizards of the Coast.