I'm not familiar with AVERY. This could be done with Inkscape, except without the downloading and storing. You'd have to download separately. Unless you know how to write computer scripts? If so, you could write a script to do the whole thing.
Are the templates in a raster ("bitmap") format, or vector? Well, actually, if all you're doing is printing, I guess it doesn't matter. There are a couple of ways to approach it.
Do you do the caricatures in a vector format? That doesn't matter either, except if you don't, then you don't need Inkscape at all.
However Inkscape can still do it.
One way:
Import the template into a new blank Inkscape file. This would be the best way if the badge template is different, each time you do it.
Other way:
If you always use the same badge template, you could create a custom Inkscape template. Or if there are say, a set of 5 or 10 that you always use, you could make a set of Inkscape templates
But unless you can write a script, or know someone who would do it for you, Inkscape can't download the badge. I don't know how to write scripts myself, or I'd offer. It's probably not very hard, for someone who knows how.
Possibly exciting, or at least interesting idea -- make your own badges/templates, with Inkscape!
Other tip -- if these badges are either licensed or copyrighted, you'd probably want to be sure before you use them.
