It depends on the country where the process is taking place. In the United States, for instance, the answer to this question would be yes, as egg donor recipients can choose a known, semi-known, or anonymous donor depending on their preferences.
In Spain, Cyprus, Ukraine, the Czech Republic, etc., donors must remain anonymous by law. In the UK, since April 2005, anonymous are not required to be anonymous in accordance with the HFE Act. All donor-conceived children have the chance to know the identity of their donor if they wish to after turning 18.
Are you interested in learning more about this? Do not miss this article: Egg donation to a family member.