I claimed a code for an Aurora cape in Minecraft, and it hasn't appeared in my inventory.

Yes, there can be a delay.

For promotional Minecraft capes, the cape typically appears in Bedrock Edition first. If it redeemed successfully but is not showing in Java Edition, use this path:

1. Open Minecraft: Bedrock Edition and check Dressing Room to confirm the Aurora cape is present there.
2. If it is present, close Bedrock and open Minecraft: Java Edition.
3. Go to Skins.
4. Highlight the current skin.
5. Select … > Edit.
6. Choose the cape and Save & Use.
7. Restart Java Edition and check again.

If the code now says it is already claimed, that indicates the redemption likely went through. Community guidance in the provided material also notes there can be delays when many players are redeeming at the same time. One accepted answer mentions a short sync delay, with reports of up to about 24 hours before the cape becomes selectable in Java.

If the cape still does not appear after checking Bedrock first and then manually selecting it in Java, the next step is to contact Minecraft Support for an account check.

References:

• Redeemed Minecraft Cape didnt show up in cape section - Microsoft Q&A
• Minecraft cape is not showing - Microsoft Q&A
• Minecraft redeemed Cape didn't Show up in my Minecraft Account - Microsoft Q&A