Yes they can
Civilians can own and use suits of Power Armor.
However, note that only Space Marines are genetically enhanced with something called Black Carapace (not related to regular "carapace armor")
that allows them to interface with their Power Armor in their unique, second-skin like way. All others (including Adeptus Sororitas and the Inquisition, although they are far from "civilians") use Power Armor as normal enhanced armor. It's still impressive, but only the Space Marines profit from the full potential.
And they might
Power Armor is hugely expensive, not available for private buyers on the regular market and will surely attract attention from people whose attention normally ends in guilty and innocents dying all the same. The Inquisition and rich Rogue Traders might be able to get hold of suits, but this is nothing that a gang of rich kids might get for their 18th birthday. Well, actually, it just might be if you go back far enough in canon. The Spyre Hunters, a gang of rich kids from the upper class did just that in the game Necromunda, where gangs fought over territory in the underhive.
However, not all "Power Armor" looks like the iconic Space Marine Power Armor Mark I to X. There is Imperial Power Armor, Ex-Imperial (aka Chaos) Power Armor, Tau power armor and probably some other variants. It goes without saying that anyone owning a non-Imperial Power Armor better be having proof they killed the previous owner and are keeping it as a trophy.