You should be able to reuse the <umb-input-multiple-text-string> component, set the value on the items property and listen for a change event to update your value.
Much like how the Multiple Textstring property-editor does it:
Hmm, I’m curious what is in the value property? I assume a string array?
Does the UI render anything at all? e.g. even the Add button?
If it doesn’t render the <umb-input-multiple-text-string> component, then it could be that it hasn’t been loaded into the browser. You could then try importing it via import '@umbraco-cms/backoffice/components';.