I know the Chromebook edition ships with coreboot but does the other models have the capability of installing custom firmware like libreboot? Does framework have to give the approval for it?
No. Framework laptops currently ship with Intel Boot Guard enabled which only allows the firmware to be updated. They have given out models with boot guard disabled to coreboot developers for them to add support. A status update at a conference was going to be given but the developer was sick and it has been delayed to next year.
Please keep discussion about Coreboot to the Coreboot wiki thread in this forum.