That is a pretty standard use case for apply():
R> vec <- 1:10
R> DF <- data.frame(start=c(1,3,5,7), end=c(2,6,7,9))
R> DF$newcol <- apply(DF,1,function(row) mean(vec[ row[1] : row[2] ] ))
R> DF
start end newcol
1 1 2 1.5
2 3 6 4.5
3 5 7 6.0
4 7 9 8.0
R>
You can also use plyr if you prefer but here is no real need to go beyond functions from base R.