Hi @Michael Williams ,
Here is the code snippet about adding Client People Picker in SharePoint Hosted app for your reference, I would suggest you to check:
In Default.aspx, Reference file under PlaceHolderMain:
145654-default.txt
In App.js, write Script below, not use this, it might cause xxx object is not defined:
145678-app.txt
Here is the document for reference
https://learn.microsoft.com/en-us/sharepoint/dev/sp-add-ins/use-the-client-side-people-picker-control-in-sharepoint-hosted-sharepoint-add-in
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