Where is the best place to farm Blood Vials in Bloodborne?

In my opinion, the safest, fastest way to acquire blood vials at this point in the game is to buy them, even though they cost 720 echoes each after fighting Rom.

Many of the blood vial farming runs suggested online are quite good, netting 15-20 vials per run; however, if you aren't careful, they'll net you much less than that, as you'll have to use some to survive against the more difficult enemies (e.g. Hemwick executioners, giant hogs, scourge beasts). You can avoid this issue by farming for echoes in a safer, more predictable environment, then buying blood vials with the echoes you've earned.

The first floor of the Lecture Building is a great blood echo farm, consisting only of Slime Scholars (a relatively non-threatening enemy), and the run will net you around 13K-15K echoes -- enough to buy 18-20 vials. The Slime Scholars also have a chance to drop Quicksilver Bullets and Sedatives, both of which are nice to have in large quantities.

To summarize, I suggest farming the first floor of the Lecture Building for blood echoes to buy vials, because:

• The run is much safer than most blood vial farming runs
• The run takes around the same amount of time to complete as a blood vial farming run
• The blood echoes you get allow you to purchase a number of blood vials comparable to that of a blood vial farming run

I understand that this technically doesn't answer the question of "Where is the best location to farm blood vials?", but it does help the asker with their stated issue, and the end result is the same.

The place I use the most for Blood Vials is in Central Yharnam. Luring the werewolves to the two-story house that leads to the bridge where you fight Cleric Beast is the lowest-risk farming location. Each werewolf should drop 2-3 Blood Vials when they do drop them (they have a high drop rate), but they cannot come into the house as they are too big. There are other enemies in the house which you should be able to one-shot at this point, and some may drop additional Blood Vials. It's worth noting that this is also quick and easy way to farm Quicksilver Bullets, as the Wheelchair Hunter drops a set 4 bullets each time you kill him and he has his back towards you when you enter the house. Overall, this route should guarantee you 4 Quicksilver Bullets and 3-6 Blood Vials (potentially more) every couple of minutes. Personally, I prefer this route due to the fact that you have virtually no risk of dying and it's extremely quick.

If you are looking for a challenging route, I might suggest Cainhurst Castle, though, according to the wikia, the recommended level is 60-80, so it may be too difficult for you at the moment. That and it will take a lot longer than the previous method. Also, if memory serves, the enemies won't drop Blood Vials too terribly often. However, the route is relatively easy and you will receive ample Blood Echoes, with which you can buy a lot of Blood Vials.

According to this wikia page, there is an alternate recommended route:

Warp to the Witches' Abode lamp and head straight back toward the first Hemwick Charnel Lane lamp killing everything along the way. The three axe guys yield about 2500 echoes apiece, and at night there are two smoke demons worth 1500 ( the second smoke demon spawns just outside the entrance to the Hemwick area). One run nets about 15k echoes with the first tier Moon rune equipped.

The good route is from Father Gascoigne fight lamp backwards, across the bridge (or trigger ball, which will fall just at enemies on bridge, and go under bridge), use shortcut elevator, and go to Cleric Beast lamp. 4 giant huntsmen, several big mutated hunters, 2 beasts, numerous common huntsmen, 1 giant hog. I'm getting around 10-15 vials per run in such route.

Giant hogs in last part of Forbidden woods are also granting big amount of vials.