Machine Api endpoint from within an App or Machine

It should work, yes (and folks have reported it working before):

For a test Machine app I run, I see it work:

fly dig AAAA _api.internal -a <app-name>

;; opcode: QUERY, status: NOERROR, id: 35964
;; flags: qr aa rd ra; QUERY: 1, ANSWER: 1, AUTHORITY: 0, ADDITIONAL: 0

;; ANSWER SECTION:
_api.internal.	5	IN	AAAA	fdaa:0:<orgid>::3

Are you querying for AAAA records? Some are TXT records too (ref docs).