Brackets that are included with this RBC036 is not a clamp type it is to slide on the C-channel of the carrier then an M6 channel nut to slide on the c-channel of the crossbar.Vehicle-Accessories
The skewer appears to be 9mm and the blocks that the skewer pinches onto the fork appear to be 18mm. Should be no problem for either 9mm or 15mm axle without any adapter.
The top of your crossbar would have a track in it. This would be accessed under the bars end caps and, often filled with a rubber strip when not in use.
If it’s a thru axle you’ll definitely need an adapter. I’ve used it for my full suspension mountain bike, road bike, track bike, and bmx with no issues (aside from adapter for thru axle in Mtb)