Of course the vampire can bite a willing target without attacking
Assuming it is happening outside of melee, at a minimum. If it is happening outside of melee it is not an attack. The willing victim is offering its neck, and the vampire is biting it.
The counterargument is absurd: The vampire slips in through the bedroom. The occupant says, "master I am here", then the DM says okay, roll initiative. It's But . . . it's not combat, there's no initiative, and the vampire doesn't have to make an attack roll. Shaking someone's hand isn't unarmed combat. Grappling a willing target outside of combat isn't an attack either, you're putting your arms around them. Putting on a bandage doesn't involve a to-hit. Picking up a kitten bumping up against your leg doesn't involve a grapple, you just pick it up.
If for some reason that doesn't convince you, rulings over rules. If having the vampire bite a willing target outside of melee seems like a nonsensical interpretation of the rules, then make a ruling. From Tasha's:
The rules of D&D cover many of the twists and turns that come up in play, but the possibilities are so vast that the rules can’t cover everything. When you encounter something that the rules don’t cover or if you’re unsure how to interpret a rule, the DM decides how to proceed, aiming for a course that brings the most enjoyment to your whole group.
In combat is a whole different thing
In combat is a different thing. The PHB says:
A typical combat encounter is a clash between two sides, a flurry of weapon swings, feints, parries, footwork, and spellcasting.
In that case, the DM will need to judge if a to-hit is needed. Perhaps the willing target is being protected by allies. An attack roll makes perfect sense.
Or, perhaps the vampire and the willing target are alone behind cover. The target is willing, and maybe even helping. The DM may judge that a to-hit is not needed, and the vampire simply succeeds.