Steven Zahm@ Margaret
In short, the answer is yes and yes.
You’ll need to install a plugin that enables shortcodes to be used in the WordPress Text Widget.
After you install it, you can use the [connections] shortcode in the Text Widget.
I highly suggest using the lock shortcode option like this:
[connections lock='true'] otherwise that shortcode instance will respond to any valid Connections query which is very likely not something you would want.
A caveat, you would have to disable the List Actions and Entry Actions on the Connections : Settings admin page under the Display tab otherwise they’ll display in the widget. Unfortunately this mean they will not be shown for any other instance of the shortcode too. That could be a show stopper.
If you use the Default Card template, there’s is an add to addressbook link, this allows the user to download a vCard. So the shortcode in the widget would look like this:
[connections lock='true' template='card']
Kind of complicated, I know. I do plan on introducing widgets to make this very simple but I do not know when at this time.
Hope that helps!