Several passages in War of the Gifts would imply that socks are worn in Battle School:
But even as a joke, it was a gift, wasn't it? Santa Claus was giving gifts all over Battle School within days.
It was more than just gifts. It was stockings. Nobody could say who started it, but after a while it seemed that the giving of every gift was accompanied by a stocking. Rolled up, hidden inside something else, but always a stocking. Nobody hung the stocking up in hopes of getting it filled, of course. It was the other way around- the stockings were being given as part of the gift.
And the recipient of the stocking found a way to wear it, whether it fit or not. Dangling from a sleeve. On a foot, but not matched with the other sock. Inside a flash suit. Sticking out of a pocket. Just for a day, the sock was worn, and then it was given back. It was the stocking more than the words now that said, This is from Santa Claus.
"I don't know yet," said Wiggin. "When I move it, it throbs."
"Bring your other leg up so I can compare ankles."
Wiggin did. Zeck pulled his shoes and socks off, despite the way Wiggin winced when he moved his left foot. The bare ankles looked exactly alike, as far as he could tell. "Doesn't look swollen."
However in Ender's Game, Ender says that they no longer wear socks:
"I want to see your fart collection," Alai said.
"I stored it in your locker. Didn't you notice?"
"I thought it was my socks."
"We don't wear socks anymore."
"Oh yeah." A reminder that they were both far from home. It took some of the fun out of having mastered a bit of navigation.
So do Battle School kids wear socks or not?