Begging for bites isn't going to get you far.
A. it's a 7 day cool people are protective.
B. many guilds already give out free bites for events or members.

The "best" way is to head to the zone where they spawn(which you need to be in to be bitten anyway as it's the only place people can bite you so you'll need to find it) get the warp point and comeback whenever it hits nightfall in game to look for NPC spawns(it can be vampires, werewolfs, or nothing they always spawn in groups of three and it can take tell midnight in game for them to spawn)

For AD it's reapers march and they spawn around dune but if your not AD people in your faction will often be happy to point you to the right zone.

Edit: You only need to be hit one time by the NPC vampires to get the virus and quest it works even if you die so no reason to kill them other then stopping others from being bitten.