Jaroslav M's excellent answer gives the key idea: certain ranges of two-digit years are (almost exclusively) safe to extend to four digits; some are "probably 19xx"; some are "probably 20xx". I don't think you should aim for no manual input, but with that concept in mind you should be able to reduce it to a minimum.
99 – Assume
For real users it is safe to assume that anyone with a recorded birth-year between
99 was born in the 20th century. The lower value,
19, must mean
1919 (despite the person now being 100) since we're still (at time of writing) in 2018. The upper value,
99, must mean
1999 (= 19 years old) since Emma Morano, the last person (known to be) born in 1899, died in 2017.
Fairly Safe Case:
09 – Assume
It should be fairly safe to assume that anyone with a recorded birth-year between
09 was born in the 21st century (and are therefore aged between 19 down to 10). Assuming
19xx would imply ages from 119 down to 109: the latter might be possible, but is very unlikely1.
Probably Safe Case:
18 – Assume
Unless you know you have records for children (as opposed to young users), then it fairly safe to assume that anyone with a recorded birth-year between
18 was born in the 20th century (and are therefore aged between 109 down to 101). Assuming
20xx would imply ages from 9 down to 1: unlikely for real users.
Verify the above
For all users you should probably display a notification (perhaps something like the Code of conduct that StackExchange is currently showing) along the lines of:
We're updating our records. Your recorded date-of-birth was dd/mm/yy which we've assumed means dd/mm/yyyy making you zz years old. Please [Click Here] if this is incorrect.
(Including the age is probably helpful, but you could remove it if you think some people might be sensitive to seeing it!)
If you do have records for children (or users under 10), then for certain recorded birth-years (perhaps
15) you might consider explicitly asking the user to confirm whether to use
19xx (ages 114 down to 104) or
20xx (ages 14 down to 4).
Similarly, if you have historic records (i.e. of people who may have died), you may want to ask a similar question (although if that were the case, you really should have been using 4-digit years from the beginning).
1 This list of oldest living people shows the 100 oldest people and notes that there are estimated to be between 150 and 600 living people who have reach an age of 110. While that's an impressive number from a longevity point-of-view, the chances of any of them using your application is probably remote.