phlebotomists are (nearly always) highly skilled people who safely and relatively painlessly poke our veins with big needles, take blood samples, then ask us to "press here". The process is called a phlebotomy.
the surgical opening or puncture of a vein in order to withdraw blood, to introduce a fluid, or (historically) when letting blood.
This requires training, practice, a medical background and will often require certification.
It's normally done in a medical setting so if there are any unexpected challenges they can be dealt with.
Phlebotomy has no doubt been common in spaceflight, medical research on the health effects of spaceflight has been central from its beginnings and is highly active today as extended missions are considered.
Question: Are all modern astronauts at least passable phlebotomists?
This would be a real challenge to do on yourself because of both geometry and human nature, every crew member will need at least one other crew member to be able to do this, so in crews of 2 or more there needs to be at least two passable phlebotomists.
Is it something that most or all astronauts are capable of doing, or are there just a few "designated drivers" in each crew?
Do astronauts phlebotomize each other during ground training for practice?
From the NASA.gov feature NASA Researchers Develop a Technique to Predict Radiation Risk on International Space Station Missions:
Japanese Aerospace Exploration Agency astronaut Akihiko Hoshide poses for a photo after undergoing a generic blood draw in the European Laboratory/Columbus Orbital Facility (COF).