I've had this problem in my old FS MTB frame, which is a 4-bar linkage.
The ones that have got play are the ones in the swing arm where it bolts onto the seat tube.
You can feel it if you stand the bike on hard ground and put a hand under the back of the saddle and very gently lift, if you lift not quite enough to lift the back wheel off the ground you can feel the slight knocking.
If they're anything like mine they are similar to the middle one in this picture
as to whether you can do it yourself, I would personally say no, largely because i would foresee that putting it back in would be hard without a special hydraulic bearing press. However I would have thought you might as well have a go, since you can make sure you don't damage it (damaging it would be achieved by say indenting the rubber internal section) and if you can't get it in you can just admit defeat and take it to a shop that can do it.
You will almost certainly have to take the swing arm off.
Make sure it actually is the one i'm talking about before taking it off - you can determine this by holding parts of the bike still and trying to feel the play by moving other parts.
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