B737 Captain reported gate departure without required Hazmat documentation. The required documents were obtained during cruise via ACARS.

Narrative

During our flight; we received a preliminary during preflight Hazmat notice over the ACARS. I made a note to send the new procedure [company] code when we got the final notice as we had done just a day prior. We got the airplane ready and did all of our checks; loaded everyone up; and took off. At cruise; I realized I never sent the code. But then we realized we never received a final Hazmat notice over the ACARS. We checked the received history and there was not one in there and nothing ever came off the printer. Initially we thought maybe they didn't load it. However we went into the ACARS -load planning; and requested a final hazmat report. It sent us the final report showing the Hazmat was boarded. At which point we did send in the code. We were parked at a gate where the ACARS was slightly intermittent so maybe that could have had something to do with not receiving the final report. We had to call ramp instead of requesting pushback on ACARS because we had a no com" momentarily. Also we could tell some of our ramp crew was in training so maybe they didn't send the report?"