Air Carrier flight crew departed a major airport VFR and picked up an IFR clearance with TRACON; due to COVID cleaning at the Control Tower.

Narrative

At completion of push FO (First Officer) called for Taxi clearance from Ground Control and there was no response. After numerous attempts we were advised Tower was closed. We then received new ATIS and proceeded to taxi to Runway XXR uncontrolled using CTAF. We had received a clearance through CPDLC. Upon reaching the runway we advised traffic we were departing XXR and commenced takeoff. Conditions were VFR. Immediately after takeoff we contacted the Departure Control listed on our clearance. After a request for IDENT ATC advised radar contact and then questioned if we had an IFR release. We stated we were VFR and requested an IFR clearance. ATC cleared us to fix and read our clearance and we acknowledged with a full read back. We proceeded on course with no further incidence. With the next Controller we ask why the Tower was closed and were told it was due to COVID and they were sanitizing the facility.There was confusion about the Tower closure. No notice had been given and we didn't realize it until we received the updated ATIS. We departed VFR and then realized later we should have gotten an IFR release on the ground from the Departure frequency or the phone number listed on the ATIS. Advanced notice from ATC to the company about the planned closure. And then company could have advised us through other sources.