when I get a notification that someone has applied to join my tribe, it would be nice to have an option to accept the application and let them in. The only option I have is to invite them to the tribe, which makes the application disappear. Is that how it's supposed to work or is something wrong there?

accepting applications
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Yes, when you approve someones Application to join your tribe, you do so by clicking 'Invite'. That is how it works.
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Moving to "Help & Questions". Yes, Furry is correct, when you accept an application you invite them to join, but it's still up to them to take on your offer afterwards. This is done so that the player can choose on his own terms when and which tribe to join, which is handy if he has for instance applied to multiple tribes at the same time. So yes, inviting a player is accepting their application.