It may be possible, but not with existing technology.
The US spent almost 2 billion dollars developing an airbreathing single-statestage-to-orbit vehicle in the 1990s. This "National Aerospace Plane" (NASP) aka the "Orient Express" project never produced any flight hardware, although a test version (the Rockwell X-30) was planned but never built.
For such a vehicle to succeed, a working scramjet engine is required. Turbojet and ramjet engines must decelerate the incoming flow to subsonic speeds in their combustors, resulting in high drag and inefficiencies. A scramjet (the first two letters of which stands for Supersonic Combustion) does not require this deceleration and therefore skips the drag penalties.
Scramjets do not look like regular engines. The engine for the NASP was basically its entire underside - it was a waverider configuration where the underbelly compressed the flow for use in the engine. While an ingenious design, no one has actually gotten this to work for more than a few seconds.
The entire possibility of airbreathing SSTO hinges on a practical, working scramjet engine. Until this technology is achieved, it cannot be done. This was the major factor in the failure of the NASP.