Using a computer, there is no way to access the text boxes in UI elements without using a mouse. Here's how to fix for each UI:
GTS: allow the up/down arrow keys to toggle between the tab and the search bar for each tab that has a search bar
Friends list: first allow the side arrow keys to change tabs like the GTS, and then allow the up/down arrow keys to toggle between the tab and search bar for each tab that has a search bar
Pokedex: highlight bulbasaur so the players know where the navigation already starts, and then allow the up/down arrow keys to toggle between the first row of pokemon and the search bar
Bag: allow the up/down arrow keys to toggle between tabs, search bar, and then tab contents in that order