The best place to get a lot of blood when you need it fresh is from someone else.
Here’s the thing, blood magic is a form of human sacrifice. The idea is a life is more powerful than drawing on or channeling the ambient energy which surrounds us in the universe at large. That is the actual magical theory. It’s sacrificial magic. You’re sacrificing your (in this case, literal) life energy in order to empower or gain the power to cast your spells. The logical choice from a survival and concentration standpoint is to use someone else’s life, especially if the spell calls for a lot of blood.
You don’t want to be losing blood in vast quantities, but you can pretty much cut anywhere to get blood. Your body is full of veins. If just a dash will do, these mages can cut pretty much anywhere so long as they don’t nick a major artery.
The problem, however, with using your own blood for spell casting is that blood loss causes dizziness and a loss of concentration. This is pretty much the perfect storm of terrible ideas when it comes to attempting to fundamentally alter reality. Reality bending is definitely the best time to lose concentration and, possibly, pass out in a dizzy haze of confusion.
In most magical systems, the magic will just keep taking if the mage leaves the spell open ended. Without safeguards, even a potentially minor spell can be deadly and cause a bleed out. That’s the problem. It’s magic, you could just bleed indefinitely in a spell gone wrong until you die. The bigger and more complex the spell, the more likely this is. If the injury won’t clot because the spell is drawing on blood inside the mage’s body, they will eventually bleed to death. It doesn’t matter where. If the volume lost exceeds the volume generated then they will die.
So, consider the operational costs. Sacrifice someone else.
That’s what cultists are for.
(And never cut across the palm of the hand. It’s not deadly in a blood loss sense, but it does suck and involves your character giving up their hand for a few weeks. The back of the hand is better, the top of the forearm, etc.)