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What a great tip, thanks! I only found that when I now put the Manipulate[] inside of a CloudDeploy[] the problem persists. However, when I save the Manipulate[] to a variable and do CloudDeploy[] on that it all works out. I hope you don't mind me changing your answer to reflect that.
@NSR as long as you place the file in the notebook's directory on your friend's computer and do SetDirectory[NotebookDirectory[]] in their notebook it should be fine.