You probably want to define a size for post thumbnails for this side-bar area first. This needs to be registered with your theme so you can refer to it later one.

Then you can make use of the get_the_post_thumbnail() function passing the ID of the post as well as your post thumbnail size name as the second argument.

Now the last thing you need to know is to get all posts of the category you like to get the thumbnails from. This can be done with a new wp_query just getting all categories posts.

Then you combine this all together.

Example Code

This is some example code to display the thumbnails registered with the sidebar-thumbnail size. You need to do this generally with your theme, probably in the themes function.php. Please see codex how to do that. This little code takes care to query all thumbnails by category and then inserts a <br> tag after every third thumbnail:

$myThumbnailSize="sidebar-thumbnail";
$myCategoryID = 12;
$myThumbnailsPerRow = 3;
query_posts('category='.$myCategoryID);
if (have_posts()) { 
    $count = 0;
    while (have_posts()) { 
        the_post();
        echo get_the_post_thumbnail($post->ID, $myThumbnailSize);
        if (0 === ++$count % $myThumbnailsPerRow) echo '<br>';
    }
}
wp_reset_query();

It’s sample code so untested, but it probably helps you to get things running. All the functions used in there are documented in worpdress codex btw, so you can learn more about them there.