Best practices for saving encrypted user data without a database
This answer proposes a solution based on local storage and local encryption.

The data of the forms that the user enters:


Can be locally encrypted whenever the user wants to save...
The encrypted file(s) can be stored locally, if you want to store it to, you can transfer a copy to your server
The encrypted file(s) can be decrypted locally and restored into forms by JScript. You can use NodeJS.
If the user lost their file(s) then if you saved in your server, you can transfer back to the user to resume.


The methods;


From a password to an encryption key, you can use PBKDF2. Ask the user to a good password to increase against the brute-force attacks. 
For encryption of the data, you can use AES-CBC with randomly generated IV. If you want to authentication for the encrypted data, you should, you can use AES-GCM.


Don't forget to ask the user to write the password at somewhere.