I'm not sure if an injector will work in this situation. In the case of a steam injector, the specific volume of the mixture is lower than that of the inlet steam, which is partly how the outlet pressure is able to increase above the inlet pressure. If you have a combustion reaction between the air and some kind of fuel, the specific volume of the reaction products will be much higher than the inlet air, so you may not be able to build up enough pressure in the outlet stream to make it work.
You may want to instead look into turbo pumps like the kind used on liquid fueled rocket motors.
If both the motive fluid and the suction fluid are air, then this will work, though the efficiency may not be as great as a steam-water injector. The inlet air will induce flow in the suction air, and the resulting outlet flow will have pressure and velocity somewhere in-between that of the inlet and suction streams. So, if the suction air is at atmospheric pressure, the outlet flow will be higher than atmospheric. Whether or not the pressure will be high enough to meet your needs depends on the pressure and flow rate of the inlet air. A pneumatic vacuum uses this same principle, you may want to look into those as well.