As announced today, this ability has been added. The plans to add this functionality were announced by FamilySearch some time ago, but it has required major changes to some of the under-lying systems of the website to make it possible.
Church Newsroom Article:
https://newsroom.churchofjesuschrist.org/article/familysearch-document-same-sex-family-relationships
Deseret Books Article:
https://www.deseret.com/utah/2019/12/10/21004733/familysearch-same-sex-family-trees-lgbtq-geneology
KSL Article:
https://www.ksl.com/article/46688581/families-can-now-document-same-sex-relationships-on-familysearch
As the articles point out, this does not represent any change to Church doctrine on marriage. But it does allow users of the website to add and document same-sex relationships.
