Honestly, Licia herself isn't necessarily all that integral to the deck, but she'll generally only cost {W}{B}{R}, and still has first strike, lifelink, and can swing as a 10/10 in her first attack, so she's not exactly a slouch.
Not vampire themed, but there are a few.
First we start out getting the ball rolling with Sol Ring, Cartographer's Hawk, Surveyor's Scope, Savai Crystal, Mardu Banner, Phyrexian Lens, Replicating Ring, Lavabrink Floodgates, and Vesper Ghoul. Blood Celebrant won't get us more mana, but fixing is good, too.
Most of this is just adding mana in our colours, but Vesper Ghoul, Phyrexian Lens, and Blood Celebrant will also let us pay life, which is useful for one of our win conditions, as well as just a thing we can do with all this mana. Lavabrink Floodgates will also potentially wipe the board, or at least keep the other players on their toes. Cartographer's Hawk is bad, but we've got ten Plains and it's the only search we've got, besides Surveyor's Scope. I thought about Burnished Hart, but iunno, at least the hawk is repeatable. Replicating Ring is just because maybe you can live the dream. Otherwise it's just a three mana rainbow rock.
Next, the Soul Sisters: Soul Warden, Daxos, Impassioned Orator, Suture Priest, Kor Celebrant, Pious Evangel, Healer of the Pride, Answered Prayers.
Aside from Soul Warden, these are only when creatures enter under your control. Soul's Attendant is over budget. Daxos will see things coming and going, Suture Priest will bleed opponents for going wide, and Healer of the Pride does double lifegain. Pious Evangel can be transformed into a Blood ArtistZulaport Cutthroat bad version of those cards. Answered Prayers dodges board wipes but otherwise is just an additional version of the effect.
Sometimes "out, out damned spot!" isn't enough, and you need to get rid of everything else as well. For that we've got Martial Coup, Hour of Devastation, possibly Lavabrink Floodgates, and Descend Upon the Fishermen.
Martial Coup, if I'm not mistaken, will have the soldiers trigger any Soul Sisters you've got, as well as Requiem Angel, but I could be wrong. It will also kill everything and leave you with a board. Descend will do even more, exiling everything except leaving you with an angel. Hour of Devastation is... well, a red spell, and like I said there's so few red pips.
Next, getting a few cards in hand. Bloodtracker, Arguel's Blood Fast, Moonlight Bargain, Necrologia. And a handful of artifacts to crack for a card draw from earlier.
Bloodtracker and Blood Fast turn life into cards, though one is a bit faster than the other. Blood Fast will likely never flip, but that's perfectly fine. Moonlight Bargain is five mana and ten life for five cards, while Necrologia is five mana and however much life you want to find the seven best cards you can, even if you have to discard to hand size.
Now for some lifegain payoffs. Marauding Blight-Priest, Cliffhaven Vampire, Epicure of Blood, Vizkopa Guildmage, Defiant Bloodlord, Serene Steward.
Since each soul sister trigger is a separate instance of lifegain, all these 1 life lost creatures will get their full value. Vizkopa Guildmage and Defiant Bloodlord will also make sure that even when we do gain more than one life in a trigger (which basically only happens with Pontiff of Blight), we still get the full value of that drain. Serene Steward isn't great since these are single instances, but the other build-your-own Ajani's Pridemates like Cradle of Vitality and Light of Promise aren't as good.
I have no idea where to put Ethereal Absolution, but it's pretty good for going wide.
And finally, the win condition (besides all the other potential ways to win): Friday the Thirteenth. To help us get to 13 life, we've got Grimclaw Bat, Bloodtracker, Arguel's Blood Fast, Reckless Assault, Martyr's Tomb, and all those others, but the best option, if we can get to the necessary 9 mana, is Kuro, Pitlord, which we can also use to mow down the rest of the board. Necrologia can also be a great way to find Triskaidekaphobia.
Otherwise, there's tokens, bleeding our opponents, or attacking with Licia and Rogue's Passage.