Steve Ray has a list of 35 loaded Questions for "Bible Christians" (quotation marks his)(link to the whole list). This is number 20/35. I'm trying to provide the answers in a common format, for easy reference.
20) How did the early Church evangelize and overthrow the Roman Empire, survive and prosper almost 350 years, without knowing for sure which books belong in the canon of Scripture?
Same as they do now, they used the books that they believed were the Word of God!
1) There was no allegedly infallible proclamation of the canon until Trent came along after the Reformation was already well established.
2) Folks were confidently stating that certain books were Scripture before even the local councils of Hippo and Carthage to which Steve Ray is obliquely referring were held.
3) We accept the Scriptures on faith (same as they did before anyone had the nerve to try to claim to be able to say infallibly which books are in and out of the canon) - not on certainty that is rooted in faith in a church.
4) And the earliest Christian writers do demonstrate that the Scriptures are exactly what they used to evangelize. Thus, for example Ignatius (died between about A.D. 98 and 117) quoted from Paul's Epistle to the Corinthians when he himself wrote to that church.