DIAGNOSIS A tiny amount of play is normal due to the oscillating design, but noticeable side-to-side wobble is not. This points to wear in the pivot pins or bushings inside the oscillating mechanism.
FIX First, rule out the blade clamp. Remove the blade and see if the play is in the metal oscillating plate itself. If the plate is solid but the whole tool head wobbles, the internal wear is the culprit. For a DIY fix, you can try applying a drop of light machine oil (like 3-in-1 oil) to the pivot points where the mechanism connects to the housing, then work the tool by hand to distribute it.
IF THAT DIDN'T WORK The bushings or pivot pins are worn beyond what lubrication can help. The entire oscillating mechanism assembly would need to be replaced. This is a moderately involved repair that requires disassembling the front end of the tool.
HEADS UP This kind of wear is accelerated by lateral pressure—pushing the tool sideways while cutting instead of straight down or forward. Let the blade cut; don't use the tool to pry or lever material apart.