Yes, it is possible to burn gaseous hydrogen (GH2) with gaseous oxygen (GOX) in the chamber. The chemical energy freed with creation of water heats the reaction products, and that energy later gets converted to the kinetic energy of the flow.<br /><br />Note that chemical energy which is freed in reaction 2H2 + O2 = 2H2O doesn't depend on the phase state of the propellants. If you add some energy to propellants before they enter the chamber and start reacting - by heating them - that added energy remains with them, adding to the resulting temperature. So, if you burn GH2 with GOX, the resulting temperature will be higher, and the thrust (per unit of reacting mass of propellants) will be higher.<br /><br />To be more precise, the increase of the temperature in the chamber shifts the equilibrium point in the burning reaction. That means, the composition of resulting gases changes with initial temperature of propellants - if you feed some plasma of H and O into chamber, they won't react much, in a sense that the resulting H2O will just as easily dissolve back. But for "normal" temperatures of liquid H2 (LH2) and O2 (LOX) versus GH2 and GOX that's a rather small adjustment.<br /><br />The volume of components does complicates things in term of construction, but doesn't reduce the thrust by itself. As an example, the Centaur's engine, of RL-10 family, feeds GH2 into chamber, after that GH2 was used to rotate the turbine which drives pumps.<br /><br />Be aware that the biggest pressure point in the whole system is immediately after pumps. Pressure gradually lowers from that point - first in the cooling jacket, then in injectors, and in the chamber it is even lower - and that is the reason why burned gases don't go back into injectors and fuel pipes.