In Scopus, you'll want to click on 'Affiliation Hierarchy' and find the top-level Affiliation ID.
Then, submit this ID through the affiliation profile retrieval API:
https://api.elsevier.com/content/affiliation/affiliation_id/xxxxxxxx?view=FULL
In this response, you'll find an identifier for a document containing the Affiliation Hierarchy:
<hierarchy-document xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<eid>xx-xx.x-xxxxxxxx</eid>
</hierarchy-document>
Take that and submit it through the same profile retrieval API, using the 'eid' endpoint:
https://api.elsevier.com/content/affiliation/eid/xx-xx.x-xxxxxxxx?view=FULL
This response contains all the children to the top level hierarchy, which you can use to search, etc:
<institution-hierarchy verified="true">
<institution xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" id="xxxxxxxx">
<institution-name>xxxxxxxx</institution-name>
<institution id="xxxxxxxx" rel-type="po">
<institution-name>xxxxxxxx</institution-name>
<institution id="xxxxxxxx" rel-type="po">
<institution-name>xxxxxxxx</institution-name>
</institution>
<institution id="xxxxxxxx" rel-type="po">
<institution-name>xxxxxxxx</institution-name>
</institution>
<institution id="xxxxxxxx" rel-type="po">
<institution-name>xxxxxxxx</institution-name>
</institution>
<institution id="xxxxxxxx" rel-type="po">
<institution-name>xxxxxxxx</institution-name>
...etc etc