Vladimir’s name day is July 15, the feast of Vladimir of Kiev.
Churches that follow the Old (Julian) Calendar keep the same feast on July 28 of the civil calendar. See everything commemorated on July 15.
The saint behind the name

Vladimir of Kiev
July 15, Saint, Βλαδίμηρος
St. Vladimir, Equal-to-the-Apostles, was Great Prince of Kiev in the tenth century. Raised in paganism, he sought out the true faith, and after his envoys beheld the divine services in Constantinople he received holy baptism and in 988 led his people to the waters of the Dnieper, the baptism of Rus.
Name days: Vladimir, Vladimiros
Also celebrating on July 15
These names share the feast, many of them other forms of the same name:
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.