Can I buy skates a little big so that my kid can grow into them?

You can buy them a little big but not too much. A half size is ok but a whole size is usually too much. If the skates are too big, their feet will move around in the skate and they won't be able to do the jumps, spins and edge work that they are learning. Weight distribution over the blade is very important for figure skating.

To spin, for example, the skater is on the curved front of the blade (not the pick) and not the the flat part of the blade. If the skate is too large for the foot, then the foot will be too far back in the skate and it will be more difficult for the skater to get their weight over the right part of the blade. This is just one example, but the precision that is required to execute the difficult moves that skaters do requires a very strong, secure connection between the skaters leg and foot and the blade. If the boot doesn't fit properly, the skater will have more difficulty.

When I was a kid, the solution to skates that were too big was to put some newspaper in the toe. This isn't going to cut it for figure skating. Sometimes, a thin insole can be added to snug up a loose fitting boot, but in general, you need a boot that fits fairly well.

A good skate shop should be able to help you out on sizing reasonably, but if your kid grows quickly, you may just have to buy two pairs a year. You may want to consider some used skates, particularly if you're near the end of the seasonand you're not planning to skate in the summer.