Can I change the email linked to my child's Apple ID?

Yes.

Here's how you change the email address (Apple ID) you use for your Apple Account while retaining everything in your account:

1. Go to https://appleid.apple.com and sign in.

2. In the Sign-In and Security section, choose Apple ID.

3. Enter the email address that you want to use as your Apple ID.

4. Choose Change Apple ID.

5. If you changed your Apple ID to a third-party email address, check your old ID email for a verification code, then enter the code. Learn what to do if you didn't get the email. --> If you didn't receive your verification or reset email - Apple Support

6. If you use services like iCloud or Messages to share with friends and colleagues, sign in to those services with your updated Apple ID. --> What to do after you change your Apple ID or password - Apple Support

For more details, read the document: "Change your Apple ID" --> Change your Apple ID - Apple Support

Same issue for me. After Step 2 above, I get a message "Parental consent is required to make this change.

An email request will be sent to your family organizer who can provide parental consent to make this change." But I don't receive an email. I've checked that the email address is correct and I've checked all boxes (inbox, junk, etc.).

Yes, see: Change your Apple ID - Apple Support