The main way to obtain mergeable items is from the Tailor of Fortune. Certain Tailor of Fortune sets are classified as mergeable and are clearly labelled like this with a green symbol in the Merge Bot interface.

During Merge Bot events, the chances of obtaining these items from the Tailor of Fortune are increased significantly.

You can also obtain special mergeable items from group competitions. The top groups from each group competition will get a very significant amount of mergeable items.

IMPORTANT: these group competition items are considered as separate from the regular Tailor of Fortune items mentioned at the top of this FAQ guide. I.e. at this stage, you cannot get them from the Tailor of Fortune like other mergeable items.