I would not use approximate dates, but you may not need any of your records. Here is a copy of the medical history form http://www.tamug.edu/corps/images/NROTCPHYSICAL.pdf
It may not be the most recient but it is very close. If you are going to answer no to every question on the form, then you do not need any of your records. If you answer yes, you will need an explanation to specific questions. You will want to know the date of treatment, what the treatment was, when it resolved and then if it ever reoccurred. So only request documentation related to answering follow up questions. You do not necessarily have to have your entire medical history to fill out the medical history form. Report based on medical diagnosis, not self diagnosis. If you doctor diagnosed something you must report. If you or your mom diagnosed something, think hard to determine if that is something that needs to be reported.
However, If you do have a medical issue, it is probably in your best interest to research the condition in relation to military service and have your records to be able to best respond.
The form does not ask about dates of immunization, dates of routine doctor visits or anything like that. Remember, if you check yes on the electronic form it will not allow you to change it to no later, it forces you to answer the follow up questions.