I'm not sure if I'm qualified. But I am a music major and I intend to go into music history for my post doctorate. Mods strike me if you don't think I should be talking here.

I'd say the standard orchestra with a wind and string section was really standardized with the Mannheim court orchestra which was the largest orchestra of the classical period (c.a. 1750 - 1815). A lot of court orchestras had a similar roster of musicians before then, but composition for a full classical symphony didn't really start until the Bach sons, von Dittersdorf, and the Stamitz family. (Symphony, sinfonia, etc. Were not clearly defined forms in the Baroque period (1608 - 1750). But typically meant that they were for a large ensemble.) The typical symphony orchestra roster includes strings (violin 1, violin 2, viola, cello, contrabass) and winds (flutes, oboe, bassoon, clarinet, French horns, trombone, and trumpets). Notice that this isn't the full orchestra that we see in the winds. Most of the brass instruments that we encounter today weren't mastered in their making until much later in the 18th and 19th century.

This is the making of the symphony orchestra outside of dramatic music. However, I would attribute the standard symphony orchestra's true origin within the opera pit orchestra. Opera originated in Italy around 1608 and was accompanied by a larger ensemble. This includes what we see in the symphony today.

I can talk more specifics if need be but I think this covers the gist of it.