Howdy Thomas,
I can see how that would be useful to embed elsewhere. While we do not have a shortcode to embed the search box, it can be easily embedded as HTML anywhere that accepts a shortcode. To do that just copy/paste the following:
<form action="/events/" method="get">
<label for="tribe-bar-search">Search</label>
<input type="text" name="tribe-bar-search" id="tribe-bar-search" value="" placeholder="Search">
<input type="submit" name="submit-bar" value="Find Events">
</form>
Would that work for your needs?
Cheers!
– Brook