Can I get a Nationals Hybrid plate for HOV 66 access?
No for two reasons:
(1) The so-called "hybrid" plate* is a particular design ("Clean Special Fuel") and you have to get that particular design.
(2) The I-66 exemption for the Clean Special Fuel plate applies only to plates issued before July 1, 2011, similar to how the I-395/I-95 exemption applies only to Clean Special Fuel plates issued before July 1, 2006. They changed the plate design in 2006 and I assume they changed it again in 2011. If you had a Clean Special Fuel plate before the date in question, you can transfer it to a new car (provided the new car is eligible for that plate) when you trade in your car, and thus keep the exemption, but if you didn't already have one, you're out of luck. For example, a couple who live down the street from us just got a new Prius and they aren't going to put a Clean Special Fuel plate on it because they aren't eligible for the I-395 or I-66 exemptions.
(So why would anyone get a Clean Special Fuel plate now? Because there's still an exemption on the Dulles Toll Road, I-64, and I-264. I don't know if the exemption applies on the right-lane HOV setup on US-1 and Washington Street through Old Town.)*"Hybrid plate" is a misnomer because other vehicles that run on alternative fuels are eligible, not just hybrids. I worked with a fellow who had a black Crown Vic that ran on compressed natural gas (the same stuff many of us use to heat our houses) and he had a Clean Special Fuel plate on it so that he could use the I-395 HOV. I knew another guy who had a CNG-powered Dodge Caravan and he had the CF plate as well; when he traded in that Caravan in 2010, he got a Camry hybrid and transferred the CF plate to that car. I've also seen CF plates on propane-powered vehicles. But people automatically assume that if the car isn't a hybrid, the person is using the plate "illegally."