Not advisable
Otherwise Bone
The answer bores down to how much blood you would need, but bone is an excellent choice. It can be made sharp, is able to handle time and by the time it's worn beyond use there is enough bone in many hunter gatherer tribes to try to make something new.
If you need only tidbits, you can suffice with a needle. It is common practice to stab a patients finger and catch the blood welling up. If this is not enough blood initially, you can get quite some millilitres extra by massaging the finger certain ways.
Keep in mind that with such technology even an innocent prick can lead to infection and death. You can of course have a ritual or primitive knowledge to have them disinfect the tools via fire beforehand, and some herbs and specific molds laying around, but in general you just don't want to be stabbed for blood. It isn't a huge chance every stab to get infected, but with many attempts and normal life adding other scratches and dirt the chances of dying before you're 20 have skyrocketed. Bone has an ok resilience against heat by the way.
If you want more like the 500ml they take at an current day blood drawing I would say you need a lot of extra people. You can draw blood with hollow needles like we do now, albeit probably bigger, the chances of death are much higher. You deprived someone not just of blood, but of a lot of other stuff as well. Iron deficiency and other complications are just around the corner. Again, in this primitive setting you do not want a single complication. Each has an significant enough chance to be part of the cause of death of someone. And this is after a single blood letting.
For collecting drawn blood you can have rocks worn down in a certain way, or possibly skulls or certain leaves if they're lucky.