Did he make his first rod in 1856, possibly, but there is no source that says CF Orvis company became a tackle company in 1856.
We're asking two different questions:
1. When did CFO establish his company?
2. When did CFO begin
making tackle?
Quite a few sources state that he turned his rod-making hobby into a rod-making business. CFO was probably in business by 1856, as his later ads and letter indicate. After all, he was recently married and needed an income. But what was the business?
The Equinox Company was established in 1853 and incorporated in 1858 "for the purpose of purchasing or building a house or houses of public entertainment, and other necessary and convenient appendages thereto..." CFO, along with at least five other Orvi, was a member of the "body politic." Could he have joined and been "in business" in 1856?
CFO listed himself only as a dentist in the 1860 census. At the same time, he managed the Manchester Hotel. June 6, 1860:

He was even busier during 1861, when he opened a new drug store that sold fishing tackle and also was "secretary" of the Equinox House. Ads from June 18, 1861, July 4, 1861, and Nov. 12, 1861:



Ads continued to demonstrate his versatility. One 1863 ad by the New Lamp Chimney Mfg. Co., 45 Fulton St., N.Y., named CFO as their agent, selling Brown's New Metal Top Lamp Chimney. Other ads, Sept. 15, 1863, and May 17, 1864:


We can safely say that CFO had been "in business" at least as early as 1858 and had started a business by 1861. Depending on how all his activities are defined, we should take his word that he was "in business" by 1856.
As far as tackle goes, we know he was
selling tackle at least as early as 1861. He advertised as early as Jan., 1871, that he was making rods. Obviously, he had begun earlier.

My take from all this stuff is that CFO
was in business by 1856, and it was his business that evolved into the C.F. Orvis Company. Note that the early advertising lacked the word "company," and I don't know when it first was used. Maybe he had made rods as a hobby way back, but there's no evidence anyone appears to have seen that proves he sold rods of his own make until later, perhaps 1870 as some sources state.
One hint we may have: George H. Swift was the only rodmaker named in Manchester in an 1870 Vermont State directory. The 1870 census named him as a "manufacturer/fishing rods." Swift later became a dentist, like CFO. A possible relationship between Swift and CFO could use investigation.