Schools, Local Chief - Bribing, Local Chief - Treaty, Improve Road Network, Improve Telegraph Network, and increasing the status of a colonial area (Protectorate -> Colony, etc.) all increase loyalty.
The problem is that the military build screen has a couple of places to the right side that look like empty slots for unit cards, which suggests (at least to me, and apparently to others) that you are currently displaying all the units you can build. Those apparently blank slots are misleading; alwa...
I am able to build missionaries and trade posts in the continental US, but I was able to reproduce your error in Samoa (I double-checked to be sure there wasn't *already* a structure in Samoa; there is not). Possibly you need to have military control? Missionaries and tradeposts are supposed to have...