We'll work with what you have
This is a family activity, we know ski gear is expensive especially when it's multiplied by 4 or more! First season, ski what you have, see if your racer is really into this before buying new equipment. Then talk to other parents on the team, kids outgrow gear in a couple of seasons, there is someone with used gear looking to trade or sell!
Here is a PDF outlining gear needed for racing at the US Ski & Snowboard level in different age groups:
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Ski boots are the interface between skier and ski, a good fit and flex is very important. There are five things to consider, boot size, toe box fit, mid foot volume, heel snugness, and cuff flex. Different boot makers use different boot "lasts," which is a form representing a foot that the boot is designed around.
When trying on boots, pull out the liner and try on the boot shell standing, slide your foot all the way to the front of the shell until toes just touch the front of the shell. Bend ankle forward and look behind the heel. About 2 adult fingers, or 25mm (1 inch) should fit between the shell and the heel. This is a comfort fit and ideal for growing kids. 1.5 fingers is a performance fit, more responsive and a great choice for that second season or feet no longer growing. The Racer fit is 1 finger, about 1/2 inch, this is quite snug, every movement counts. This tends to be more FIS racing and beyond. Now that you have a boot length, get the size as well as shell length +/- 5mm, it helps when trying on other boots.
Put the liner back in the boot and try it on, buckling it up, check flex. Not the number used on the boot model, can you move the cuff with your shin? With toes against a wall, can you flex forward until your knee touches the same wall without your heel coming up? If the heel comes up inside the boot, tighten the mid-foot buckle, and try flexing again. If the mid-foot buckle is as tight as it goes and your heel still moves, mid foot volume is too high, not the boot for you. If the heel is contained, can you flex knee to wall? If your heel is down, its easy to flex the boot and your knee is past your toes and the toe lug as you flex into it, its flex is too soft. With heel down, flex forward knee to only the toe buckle or less, it's too stiff. Error towards softer rather than stiffer, room temperature flex is similar to flexing those boots in outside winter temps on skis. The ski leverage is needed to over come cold, stiff shells.
With flex, heel, and mid foot sorted out, time to consider toe box. Now that the heel is back and snug, with good shin pressure on the cuff, can you wiggle your toes or are a few jammed together? Custom boot work can be done but for growing feet, might try another boot if toes are tight.
Kids and some Junior ski boots have toe and heel lugs that are kid sized (Type C). Somewhere between size 4 - 6 (21.5 - 23.5) they change to adult sized lugs (Type A). Check the boot centerline area on the side of the sole. A Type C binding has a DIN maximum of 4.5 or 7, whereas a Type A binding has a DIN of 7, 9 ,11 or higher, even if its labeled as a Junior binding. A Type C boot will shift left and right or up and down in a Type A binding, few bindings accommodate both A and C boot lugs.