First go to the mod folder:

\Steam\steamapps\common\Don't Starve Together\mods\workshop-1180681495\

Now open the file modmain.lua and search the lines:

local button_order = {2,1,2,3,4,5,6,3,4,1}
local button_order_scythe = {2,1,3,4,5,6,7,3,4,1,2}
local button_side = {1,0,0,0,0,0,0,1,1,1}
local button_side_scythe = {1,0,0,0,0,0,0,1,1,1,0}

This lines determine the order of the buttons and the side.

• The side 0 represent the left side, where the tools are usually. And the side 1 represents the side right of the weapons and armors.
  
• The order starts in 1 for the both sides separately, so the numbers are repeated in that arrays.

Each position of the arrays correspond with the keybind in the order of these array:

local KEYS = {
KEY_WEAPON,
KEY_AXE,
KEY_PICKAXE,
KEY_SHOVEL,
KEY_HAMMER,
KEY_PITCHFORK,
KEY_LIGHT,
KEY_ARMOR,
KEY_HELMET,
KEY_CANE,
KEY_SCYTHE
}

So for example if you change:

local button_order = {2,1,2,3,4,5,6,3,4,1}

for

local button_order = {2,2,1,3,4,5,6,3,4,1}

You swap the axe button with the pickaxe button.

*Note: remember that the buttons back is not dinamic, so you can't be able to put in more than 6 buttons on the left side and more than 4 in the right side, without it seems bad.