Yes, this is just as possible as a jet dragon, though rather more limited in range due to the need to store both oxidiser and fuel. The most efficient form of rocket engine would be a liquid-fuelled type. Then, as this is a biological rocket, the oxidiser and fuel would best be HCNO-derived chemicals, those being the most readily available to carbon-based organisms.
Given the requirement for HCNO-derived chemicals for both oxidiser and fuel, and that we must presuppose that these fuels are not externally obtained and must be synthesised by the organism. Of the many fuels and oxidisers, Hydrogen peroxide and Monomethylhydrazine are probably the most likely combination, as both are liquid at body temperature and standard air pressure, making them easy to store, and both can be biosynthesised. I don't believe this combination is hypergolic, but providing an ignition source such as an electric arc would be feasible.
Storing these fuels would be a minor issue, since they are ordinarily toxic, but it has been shown that animals are capable of storing toxic substances safely.
Reaction Chamber and Exhaust:
The reaction chamber must be strong and temperature resistant. There exists the possibility that our rocket dragon could be able to precipitate metals or graphite sheets that would allow a strong, temperature-resistant combustion chamber and nozzle to be formed. The reactants could be passed through the walls of the combustion chamber prior to mixing to both pre-heat them and cool the chamber walls, preventing excessive heating of the creature's body. Provision of other insulation would reduce further heat transfer, as would strategic positioning of the organ so that it would not dump its waste heat directly into the creature's core body. The combustion chamber might best be located at the end of a short, strong tail. The concept of a combustion chamber like an extra anus and co-located with said orifice would lead to heat management issues in much shorter order.
Since the reactants involved can be and are biosynthesised in various organisms, and have benefits other than propellants (such as protective toxicity in the case of MMH), they could occur. Then, if, like a bombardier beetle, this organism evolves a defensive heat source on its rear end, it is not too much of a stretch to progress to using that to assist flight by adding extra reactants. This system would have the added advantage that it would not only provide a means of rapid escape from one or more predators, but would also be highly likely to injure or kill at least one predator if triggered at close range.
As to size, the most efficient, manoeuvrable flyers are small, so these rocket dragons would most likely be in the 6-foot range specified in the question, and quite lightly built, save for a large propellant-storing belly. However, larger fliers have less parasitic drag, and induced drag decreases with speed, so a larger size is not out of the question, especially as rocket assisted take-off is a given.
The duration of the rocket burn would be dependent on the power of the thrust. It may be possible to maintain sufficient thrust to maintain altitude for some minutes, or to expend the available reactants in a few seconds to provide a rapid getaway or increase in altitude.
The main issue would be the synthesis of the reactants. It would take significant energy input and time to synthesise the reactants, so the reactants for the minimum six-second full-power burst might take days and a very high energy input to synthesise. This would preclude any herbivorous lifestyle. However, having a rocket would make for a predator that could not be outrun, only outmanoeuvred.