I'm going to give a bit more insights, you are on the right path.
The final step is to add, as you said, the default Contact notes Civi has for each contact, the notes field is a Multiple record set field, which as it's own name says allows to create multiple data entries for the same set of fields, ie. you can have 10 notes entries, so every time a user submits the form it will create a new note instead of updating a existing one. Please note that the Multiple record field is only available for the Contacts entity, also make sure that the profile is set to update in the settings accordion.
If you don't want to use the existing Notes that Civi provides then you could create a new set of custom fields, navigating to Administer - Customize data and screens - Custom fields - Add set of custom fields give it the desired name (for example Enquiry), Used for -> select Contacts once done a checkbox will appear saying something like Does this custom set allow multiple records make sure you check it. You could display it as a Tab and add a custom field like Subject (Alphanumeric - Text) and a second field Message/Enquiry (Note - Text/Rich text). Once finished you can add the created custom fields to your profile. If I didn't explained myself you can always have a look at CiviCRM documentation or leave a comment here.
PS: if you go the custom field path, you will also have to give the Access all custom data permission to anonymous users if applies, User and Permission - Permission (Access control) - Wordpress Access Control