According to the FAQ, yes.

"All of the microbial sequencing data and health and lifestyle information that we collect are made public through the European Bioinformatics Institute, enabling researchers from all over the world to ask groundbreaking questions using the American Gut dataset."

Yes. They even provide the Jupyter notebooks that power all their analyses so you can re-process and re-analyze the data. I was looking into AGP recently because I had the idea that variance component estimation could be used to quantify how much gut bacteria composition matters, along the lines of GCTA for genes, and they were by far the most open and best dataset around.