Can 16S V4 amplicon sequencing data be used to construct a phylogenetic tree to illustrate the evolutionary relationships between organisms in a microbiome?
If so, what is the best method to do it, starting from an ASV table?
You can use any pipeline you want to use to generate the sequences. But as you are referring to ASVs, I hope you really understand the concept of ASVs and the pipelines which produces ASVs, as qiime and mothur do not produce ASVs (however qiime2 can).
You just need to extract the sequences, align them and make a phylo tree. It is not different from any other gene or phylogenetic tree construction.