Answer: There is little potential need for venipuncture during spaceflight, so “poor veins” would be a poor reason to reject an otherwise qualified astronaut candidate. Why do you think easy IV access would be a qualification requirement for astronauts?
There are 2 reasons for needle access to veins:
Diagnostic. Chemistry diagnostics (e.g.: glucose level) can usually be done be done with capillary blood (finger poke) so veins are not involved. Some hematology diagnostics require whole blood with intact cells (e.g.: differential white cell count) and so samples are usually obtained by venipuncture (phlebotomy). But using capillary (finger stick) blood gives reasonably accurate results with an automated analyzer. Bottom line: any diagnostic blood samples needed on a space voyage could be provided without venipuncture.
Therapeutics . Fluids and medications can be delivered through a number of non-oral routes:
Transcutanious (across intact skin), and sublingual (under the tongue). This works well for fat soluble medications such as Fentanyl, Lorazepam and Nitroglycerin.
Subcutaneous. A short, fine needle is used to deposit medication in the fat layer just under the surface of the skin. Administration is simple and requires no training. Medicine is slowly absorbed into the capillaries, then carried by the circulatory system. Local anesthetics and Morphine can be administered by this route.
Intramuscular. Medication is injected into a muscle (usually the shoulder) using a longer needle than for subcutaneous. Minimal training is required. Most antibiotics can be administered intramuscularly.
Intravenous (directly into a vein). This is required for medications which require dilution to avoid tissue irritation. An example is Vancomycin. It is also useful for replacing large fluid volumes such as in trauma resuscitation. Volume replacement is particularly urgent in hypovolemic shock (e.g.: massive blood loss) but IV access can be a significant problem due to peripheral vein collapse.
Fluid Replacement via Rectum (FRVR). This is an alternative to intravenous fluid replacement when IV access is not available. It is not routinely used in emergency departments due to availability of intraosseous (needle into bone marrow cavity) and peritoneal (tube into abdominal cavity) methods. Space medicine is a particularly good application for FRVR since it requires virtually no training and astronauts are on a perpetual low residue diet.
So phlebotomy is unlikely to be an essential procedure in space. Your suggestion of using arterial puncture is not an alternative. Arteries are deep, high pressure and have thick walls. They are technically more difficult to cannulate than veins, and shouldn’t be used for administration of medication. Due to high pressure, misadventure during microgravity arterial puncture results in excessive excitement for all involved.