No. You are not permitted to ask participants for any personally identifiable information like emails, phone numbers, or addresses.
However, you can ask for participants' Prolific email addresses, which will look like firstname.lastname@example.org and can be used like a regular email to run multi part studies that require an email. Please read more about this feature here before utilising it in your study.
Please note that we have a anonymous messaging system in place that allows you to contact participants. Also, for longitudinal studies, you can invite participants to your follow-up studies by using our White List feature.