As for your original question:
Can I use the
lightning:fileUpload component with a Customer Community user?
The answer is - Yes, you can use
lightning:fileUpload component in Communities.
As for the error you are getting, you may like to verify if you are uploading the file to the correct record or not and that if you have the right access on the object/record provided for the community user to be able to upload the file. E.g., if you are uploading a file to Opportunity in a Customer Community, you will get error exactly as you have reported.
Below screenshots are from my test community, where I was able to upload a file successfully to Contact, whereas I got exactly the same error as you reported, if I uploaded it on an Opportunity.
Failed to upload (when tried to upload on object not supported in the community)