Is there Biblical support for basing your beliefs on those who've come before?
Yes. Think about how much the New Testament references the Old Testament -- all over the place! Here is just one of many sites that list New Testament references to Old Testament prophets.
The same could be said of any Old Testament prophet who referenced a previous Old Testament prophet. All of the prophets after Moses referenced the Law revealed by God through him.
This doesn't really answer the second part of your question, but serves to illustrate the process of unfolding revelation over time that occurred from Adam to Enoch, Noah, Abraham, Melchizedek, Moses, Elijah, John the Baptist, Christ and the apostles. The need for a renewal of revelation was due to the rejection by the people of previous prophets. Further, each prophet gave specific guidance and counsel for their time, as when Isaiah advised kings in Jerusalem on matters of war and politics.
Latter-Day Saints believe this pattern has continued -- or rather, that God has continued this pattern -- to our day, calling new prophets/apostles in the same manner and for the same purpose as before: to renew and reaffirm and clarify previous revelations and also give timely guidance for the current time. (I think that answers your second question about a denomination that uses non-Biblical sources).