All A-Mount lenses, all the way back to Minolta's original offerings from 1985, are compatible with all A-Mount bodies, all the way up to the Sony A99 II.
The only exception is that some very early Sigma A-Mount lenses may not work. Instead of buying a license to the A-Mount, Sigma chose to reverse engineer it, so older lenses may not work on newer bodies. Sigma offers free updates for their more recent lenses when this occurs, but that offer doesn't extend back to antiquity (relatively speaking). For instance, I got ahold of a circa 1988 Sigma 75-200 mm f/2.8-3.5, that wouldn't work when mounted directly to an A-Mount camera, but worked just fine when mounted through a generic 1.4X teleconverter. Sigma couldn't update that lens.
So there's no reason to expect that the Sony 55-300 would have any problem when mounted on a Sony A700.
¹ The Lens is the Thing ² 'Full Frame' is the new 'Medium Format' ³ "One good test is worth a thousand expert opinions." - Tex Johnston, Boeing 707 test pilot.