That requires a new board on that side, it's not repairable. That's a stress area on a bed, any attempt to repair it will fail quickly with use. My guess is you tried to move the bed on the carpet by pushing it, rather than lifting it, causing that blowout? That kind of joinery method cannot tolerate excessive side-loading stresses.

Cut a small section of 2x4 blocking and glue and screw it to the frame. Screw the broken section to the top of the blocking. Easy, strong, and doesn't have to be pretty.

Can’t put a new board on that side the side is designed with fabric to match the bed and I can’t get a replacement from the manufacturer

Find a piece of steel the width of the broken board long enough to extend past the broken part of the board. Bend a 1 1/2" to 2" section on one end at a 90 degree angle to angle down. Drill holes in the bent section and long section to allow bolts to pass through. Bolt the long section to the broken board and drill and bolt the headboard to the angled section.

I don't see a way to fix the broken board that will hold up, it's junk. You could replace the wood with Iron rails with legs used in real beds and attach the head board to it along with the foot board and sides if you can find one, they can be found for free in many cases.

It would be fiddly work but it could be fixed.

Glue the splinters back into place w/ Titebond, using clamps. Sand smooth& flat, removing all the finish up to the first cross board. Cut a short section of 1x lumber--hardwood, like maple--to the top of the injured board. (Drill holes in the hardwood splint first so you can place the inserts into the wood.) Glue and clamp the splint to the repaired board.

I would put splints on the other three corners while you're at it to prevent them from splintering out like the one corner did. Be sure to sand all the finish off so you get a good glue bond.

Wouldn't be a bad idea to splint both the top and bottom of the long boards at all four corners, sandwiching the original board between two pieces of hardwood.

For extra measure, you could take most of the stress off by adding cabinet legs--three on both long sides.