Looks like the insides are finished, right? My first choice would be is the re a way to glue to the side, maybe with epoxy?
If that's not an option since they are painted you could screw through the front face frame, countersink, fill and paint over.
Or maybe screw at an angle through the edge where the door wI'll cover it.
At the back you could screw through the side, c/s fill and paint. I don't know how thick the sides are but even if you couldn't t c/s it wouldn't be noticeable.
If none of these are options you could line the insides with a piece of 3/4 ply and attach with glue/screws/glue blocks. Mount slide spacers to 3/4.
There are rear mount brackets available for drawer slides. If the back is too thin, you could attachť a strip of 3/4 ply as just describes and screw through edge of face frame as described before.
Hope this helps.