Check spam folders?
Can you see if the account was actually created? https://www.chess.com/member/
Use that link along with the chosen username and see if it exists and is a new account.
If you can't find the e-mail in spam/junk or the account wasn't created, you can open a ticket and staff can look into it:
https://support.chess.com/customer/portal/emails/new
Hi!
My wife is trying to create an account on chess.com (desktop version) and after submitting a username and an email - nothing happens and she doesn't get any email - what could be the problem?