Unfortunately, there doesn't currently seem to be a way to do that without using the mouse. Fortunately, you can work around this by not using the search feature. If you're looking for a specific item in your inventory, don't forget that you can sort your inventory. I have had to deal with this issue, as I am playing this game on a laptop that has a trackpad (which is awkward to use) instead of a physical mouse, but it fortunately doesn't ruin the game.
On the other hand, you might find something like this to be useful in this sort of situation: https://help.ubuntu.com/stable/ubuntu-help/mouse-mousekeys.html.en
(This post assumes that you do not have access to a mouse, as I saw you mention in a topic on the General Discussion forum.)
(If any Tangledeep developers see this post, please consider this a bug, and please fix it so that one can exit search mode by pressing Tab again, and then use the keyboard to navigate your inventory. Also, this ought to work consistently on all dialogs that have a search box, as currently different ones behave differently (and never correctly).)