As chronicled in the character's comic book history compiled on the Marvel fan wiki, Moon Knight has indeed been a member of various Avengers teams.
Perhaps Moon Knight's most notable team-up was with the West Coast Avengers, which was founded by Vision to supplement the mainline Avengers during the Secret Wars event. Other notable West Coast Avengers members recognizable to MCU fans include Iron Man, Ant-Man, and Hawkeye. Moon Knight ultimately aided and joined the team after they were temporarily transported to Ancient Egypt.
Later on, Steve Rogers recruited Moon Knight to be a member of another Avengers spinoff he formed called the Secret Avengers, alongside characters like Black Widow, Beast and War Machine. Finally, Moon Knight worked as a sort of arm of the Avengers during a stay in Los Angeles, in addition to siding with the Avengers during a conflict with the X-Men.
With regards to the MCU, then, plenty of precedent exists for Moon Knight to team up with some of its most notable characters, be it as a West Coast Avenger, a Secret Avenger, or even as a plain old Avenger untethered to his comic book source material.