I'm not aware of a solution that matches what you're ask.
I see you're using field-list appearance to group all the vaccination questions on one screen. Is this because you're using tablets with large screens? Because the data collectors may scroll up and down the list to enter the data based on the order of list on the vaccine card? Some other reason?
Would you consider breaking it out onto separate screens, creating two questions for each, and requiring all questions? For each vaccine you can have a select_one question that asks: (1) Date recorded and legible (2) No date recorded (3) Implausible date recorded (4) No vaccination recorded.
You can add autoadvance (appearance=quick) the question so that the interaction is a bit faster/easier. And then it only shows the date question on the next screen if they answer 1 (and 3 if you're interested in collecting the implausible date).
I would be cautious of using instructions such as "Leave the vaccination date blank if..." since you don't know if they skipped on purpose or on accident (and if on purpose, you don't know the reason). I see you currently have a "Why was no date entered?" question that shows if the date was blank, but you don't have that required either so they could easily skip that as well (and their free text response will require extra work to analyze).