Is there a website that offers current LIBOR rates for all tenors for free in machine readable formats?
It depends a little what you mean by "current" but the CDS market developed a "standardized model" for transforming between upfront and spread-based quotes. The model depends on an agreed curve of risk-free rates.
The LIBOR rates used for CDS settlement are available at:
This is not up-to-the minute, so maybe it's too slow for you. I'm not sure what the terms of service are, but they must be reasonably liberal in order to allow the industry to trade off them.