A single sided board in Fritzing will do this, you would need to be careful to insert the led the correct direction (I don’t think you will be able to change the LED to bottom of the board in Inspector on a single sided board, but try it!) However soldering it will likely be a problem as the hole isn’t plated through and the LED body will be in the way to solder it I think.
The holes are not plated through, but it is possible (with care and practice) to do the soldering from the non-copper side of the board. The component lead will ‘wick’ the solder through the hole, and connect to the copper. But since you can not “see” that, you have to gauge success by how much solder is used, and flows into the hole. Tricky, error prone, but possible.