I'm not using UEX, I'm using UltraEdit for Windows and therefore can't help with verified information. On Windows switching between tabs within a multi-tab dialog can be done with Ctrl+Tab (left to right) and Ctrl+Shift+Tab (right to left). Perhaps such a general key switches also the tabs in UEX.
If the IDM developers have not set accelerators for the various options in the Find/Replace dialogs, please request that by email to IDM support.
On UltraEdit for Windows the situation with tab sequence and accelerator keys was similar once in the past. So another user and I also mainly using keyboard and only rarely using mouse requested a better support for keyboard users for all 6 Find/Replace dialogs (standard Find/Replace, Find/Replace in hex editing mode, Find/Replace in Files) by email and we got what was requested in next released version. I wrote in my email for every dialog the tab sequence I wanted and added for all 6 dialogs screen shots with red marked those letters which should be accessible by key.
If no user complains about missing accelerator keys and making a suggestion, nothing will change. Posting here in the user-to-user forum such a request is without an effect as this post is most likely not read by IDM, see top of this forum page. You don't need to be already a licensed user for requesting and suggesting something UEX related by email to IDM support.