B737-700 flight crew reported communication equipment problems following pilot illness incident.

Narrative

Aircraft X started out as a normal; uneventful departure and climb. At around 35;000 ft on climb out; and prior to cruise; the FO (First Officer) asked if he could go back to the lavatory. I called back to the a FA and got his restroom break set up. While waiting for the 'ok to come out' callback from the a FA; the FO vomited all over the center control pedestal and my right side. After a few seconds to realize what had just happened; I made sure the aircraft was still operating normally (Autopilot B and a VNAV level off to cruise at 37;000 ft). The FO was obviously terribly embarrassed; but he was able to act as my Pilot Monitoring for the rest of the flight. He was not incapacitated; but I instructed him that I would take over as Pilot Flying. I wanted him to just be a very thorough PM (pilot monitoring) for the rest of the flight to back me up with my additional tasks.I then called the A FA again to let them know what had happened; and that we would need some additional galley supplies for an attempted clean up. While attempting to use the intercom; we realized that the entire communications system (VHF 1; VHF 2; intercom; both CA (Captain) and FOs Audio Control panels; headsets; and hand mics) was either failed or working incorrectly. One of the radio transmitters developed an intermittent stuck mike; yet none of the microphone transmit keys anywhere in the cockpit were being pressed. We were unable to transmit on either VHF 1 or 2; but could intermittently receive and transmit on ATIS on VHF 2. We attempted every possible transmitter/receiver combination we could think of; calls in the blind; etc. While I was doing that; the FO was in the Quick Reference Handbook reviewing to see if anything was directly applicable in the Communications or General sections. Since this was not working; I decided to use my authority to ensure a safe outcome. I set the transponder; and sent Dispatch an ACARS briefly explaining what was going on. I asked them for priority on our behalf; and advised we would continue to ZZZ via the filed route and fly the RNAV RNP M with our ETA. Amidst all this going on; the FO mentioned trying to put the Audio Control Panel into Alternate Mode (which forces VHF 1 to the CA side only; disables interphone; etc). When I put the CAs ACP (Audio Control Panel) into Alternate Mode; I was able to regain intermittent transmit ability on VHF 1. We then contacted Center; explained what was going on; and advised them of our plan in the event we lost communications further along in the flight. I then called the flight attendants; explained what was going on; and did a test briefing at the end to make sure we were all on the same page. I then made 2 PAs to the cabin. The first was the standard 'prior to descent/goodbye PA'; the second a few minutes later was an explanation that they could possibly see some priority vehicles near the aircraft on arrival. I simply explained that we had a radio malfunction; but the aircraft was safe and that we would be landing shortly. The FO and I then completed all the normal approach checklists and procedures. The descent; approach; and landing were routine and safe. Since VHF 1 was working intermittently; I was able to get a handoff from Center to ZZZ Tower. I told Tower we would be unable to make any of the ZZZ Channel Traffic calls; so he asked another plane to do it for us and kept the GA traffic off the final approach course for Runway 8. We arrived at the gate; and the FO was met by ZZZ paramedics for evaluation. I spoke with the ZZZ gate agents; and then wrote up the inoperative equipment in the logbook for ZZZ maintenance. I then called Dispatch; the FODO (Flight Operations Duty Officer); and completed this required report.