You can. Locate a Class III dealer or a holder of a manufacturers license. They will be happy to guide you through the process, paperwork, provide generic CLEO form letters and information regarding license fees. My friend has a registered/licensed VP-70 by HK. It is fun to rip through the magazines! I suggest a trip to KY and Knob Creek particularly.
Sorry, but post-May 1986 machinegun conversions are illegal for civilains to own. The only way to get a full auto AR/M16 is pay around $12,000-$15,000 for a registered DIAS/Lightning Link or $15,000+ for an actual M16.