the last time i used EDI/HIPAA over the Web (5+ years ago?) it was via XML. There are a number libraries and tools to help this along.
IIRC, most EDI interactions are "file-transfer" style and don't need much interaction information (e.g. no affordances inline).
i suspect, if i was on an EDI project today, i'd be using the "representor"[1] implementation pattern to get clear separation between the internal models and the representation format and then use any of the standard formats for Web client interaction (HAL, Cj, Siren, Mason, UBER, etc.) and offer EDI/EXI-XML in responses for consumers expecting that format.
IOW, treat EDI (or EDI-XML, etc.) as just another format, not a protocol or full-stack implementation constraint.