After you checkout from KSP Store, the system will have automatically create a KSP Store Account with the email address and password you type in the Checkout Form. As of August 31st, this account has been upgraded to a Private Division account, but you may be required to complete the upgrade by entering some additional information to verify your eligibility.
You should be able to log in using those details and the instructions here. If you do not receive a confirmation email, due to your email provider blocking it, don’t worry, you can still log in and download the game.
If you are still having issues, please submit a ticket here with any details and we'll look into it for you.