I can confirm/verify that many (50+ times, 25 different people wearing medical coats with company logo) of my phlebotomists regularly advised "Use Vitamin E because it prevents scar tissue." verbatim (when sitting for Vitals Check, &/or during blood plasma donation process, often repeated by docs/nurses/staff if I go the full twice a week [U.S. Food & Drug Administration {FDA} max allowed by law is 2 donations per week], especially if you've a noticable mark forming/developing, generally seems a kind suggestion).
If you're referring to the iodine er alcohol based disinfectants (to clean the area) they're applying 30 seconds before inserting the venipuncture needle, I never heard or was told that is linked/connected/effected by or with Vitamin E specifically, though of course the cleaning effect is the goal to sterilize (which would theoretically clean away the E), I think generally the advice/suggestion is just to (re)apply Vit E regularly and keep checking results to find the correct self-dosage. I've been mainly told to apply to the skin, but occassionally they'll suggest taking a few drops orally as well (as a precaution maybe, or to balance if you do not get enough from food), though direct epidermal application is the primary method.
From personal experience (about a hundred donations total so far), I never bought any (ever) item from the pharmacies' alternative medicines isle before blood plasma donation, and then I just had to buy another vial of Vitamin-E because when I stopped the scars came back. Before now I only believed in prescription medications, so Vitamin E is impressive. Pharmacies have the concentrated form of Vitamin E which is usually what is told to find for medical purposes.
All said, it is interesting that there is not any clear signage suggesting Vitamin E in donation centers' heavy promotional and medical educational materials which are usually filled with pretty varied risk assessments and guidelines ("Drink lots of water, eat a good meal." in funky print, or on the PA system reminder between hip/pop songs), and yet all (easily 90%, meaning the questions must be frequent about that ugly scar tissue) of the phlebotomists I've worked with knew to suggest the product (there could be a good very extremely profitable cross-promotion opportunity between donation centers and Vitamin E producers to be made), so I gather there is some formal and informal uncertaintly in line with https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3486820/ (http://web.archive.org/web/20201011135835/https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3486820/ , https://scholar.google.com/scholar?cites=18050247519683909952&as_sdt=20000005&sciodt=0,21&hl=en), generally speaking about the greater blood donation world, saying "It is time to shift attention back to donor health, which is indeed no less important than insuring a safe blood supply." ...because medical science is not there yet, there is not any official word yet.
I link to the US National Institutes of Health generally there because there is not any specific known Vitamin E + Blood Vessel information listed, nor for FDA guidelines apparently.
What does it mean if that the phlebotomists/doctors/nurses/staff make suggestions (for Vitamin E, pronouncedly) but the organizations behind them do not want to speak so clearly (pronouncedly about Vitamin E)? For us it means we might have to make the decision on our own about what to value (in terms of Vitamin E).
In terms of your question wording--I notice you wrote "after" in the question title, but you wrote "before" in the question body--do you mean Before + After then? I asked & was told to apply afterwards, you were told before? So if regular, both?
I apply before and after between several times a week and sometimes multiple times a day depending on the severity of the wound, which depends on who the phlebotamist is that sticks you, and the angle of the needle maybe, which varies because they're human (and if you by yourself move your muscles while needled it can tear, you're human too), so you need to be the final judge of how much you were pricked and how it's healing.