How to allow survey participants to contact me without breaching anonymity?
This answer may depend on how many participants your survey has in general - but I think one of the main objectives of conducting a survey "remotely", online, rather than personally, in a controlled environment, is to gather an exceptionally large amount of participants. Therefore, I'll assume a sufficiently large number.

Note that online surveys are quite an exception here in that you never get into contact with the participants. In general, you always know to some extent who participated in the survey (hardly anything else is possible when you have to meet in person with the participants). And even for online surveys, contact details are often required if there is any compensation for the effort of taking part, or only the offer to be notified once the results of the study are ready (which is generally the least that should be offered to participants in response to the time they reserved for you).

Therefore, the anonymity assured to participants generally does not mean that researchers never learn about any of the personal data of the participants; it means:


Responses will not be connected to the personal data of participants.
The personal data will never be published in detail, just as aggregate values (e.g. age minimum, maximum and median across all subjects).
Any identifying data (e-mail addresses etc.) will be erased as soon as possible.


Unless there is just a small total number of participants, participants can feel reasonably safe that even though the researchers know both their personal data and their responses, those two won't be connected in the future.