This would be a programming job, and there would be a fair bit of it too. Nothing major, just lots of tentacles.
The main changes would be to the create_account template where the input form would need changing to ask for day and month instead of date; the equivalent module file whose validation of the DOB would need to change and the database write further down for which a year would still be needed because of the database structure, but could be a default year that you use for all customers; the javascript that validates the page would also need to be changed and the language file entries that give an example of the date format and the error messages would need amendment.
Then with those changes out of the way you would need to track down all other places where the DOB is used. The customer page in admin and the account edit in the catalog are two that spring to mind. And a quick search of a Zen Cart release package suggests other possibilities that may or may not need to be amended.
I would rate this as needing an intermediate level of Zen Cart and PHP programming skill.
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