DIAGNOSIS The most likely suspect is a failed trigger switch or a break in the wiring from the battery terminals to the switch. These components live in the handle and can fail from dust ingress or just wear and tear.
FIX Do a quick external check. Look inside the battery port on the tool for any obvious debris or bent/dirty terminals. Give it a blast of compressed air. Try inserting and removing the battery a few times to scrape any oxidation off the contacts. Sometimes it's just a bad connection.
IF THAT DIDN'T WORK It's almost certainly an internal electronic fault. The trigger assembly or the main control board has likely died. There's no user-serviceable fix here; the tool needs to be diagnosed and repaired at a DeWalt service center.
HEADS UP Before you ship it off, make absolutely sure it's not the battery. Even if the battery works in another tool, try a second, known-good battery in the oscillating tool. Battery-to-tool communication can sometimes be finicky.