a standard machine can sew webbing. just slow down and use a fat needle.
for the rest of your question, i do not know. if i am i understanding you correctly, my approach would be to fold the three sides, turn it right side out, then fold the fourth edge twice to hide the raw edge, and sew it. that edge would be the edge you can't see when the sling is in use.
Thanks. I now understand what you mean by sewing the final edge in that manner. I suppose it all comes down to personal preference as to whether you stitch is that way or use a grosgrain to hide the raw edges.