True, but generally they were much better positioned to do it, for the UK it would have meant an amphibious assault at the end of very long supply lines, not quite as bad as the crusades perhaps but still... intensely problematic
If you are still speaking of the medieval era, the idea of supply lines is very anachronistic. Most of the time, armies lived off of the countryside. This continued to be the case up until around the dawn of early industrialized warfare, and even then, such as in the American civil war, many armies lived off of the land, the most famous incident of this being Sherman's March to the Sea.
It's anachronistic in some senses, but there are still issues to be considered: reinforcements, replacement equipment and what the hell you're going to do with the army when winter comes and so forth, some of which can be constructed in the field or repaired, others can't: a skilled fletcher can fetch arrows and a bowyer can indeed work from what is available (although some woods are better than others) but replacing cannon, repairing swords and armour and so forth needs smelted metal, some of that you can bring with you, you might be able to capture more but if you're setting up to take and hold vatican city you're going to need regular supplies for garrison forces, pillaging the countryside isn't capable of supporting a large army(which you would need since the Catholic nations wouldn't take a vatican invasion lying down (aside from anything else it mucks up *their* plans for controlling the papacy) for prolonged periods of time since sooner or later you've stripped the area raw: the taking isn't the hard part, as you've pointed out an army can live off the land to do that, it's the *holding* where the pseudo-supply-lines start to come into play unless you want to turn the whole thing into a full-on invasion of Italy (which if going after the vatican alone hadn't got spain and france to put their differences on hold, this would) since the vatican itself is too small to be really self supporting of more than a minimal population so either you end up with scope creep where you end up taking more of Italy, probably trigerring a larger pushback which ultimately Britain couldn't have won except maybe at the height of the empire, not against France, Spain and Italy combined or you have to look to bringing in supplies from elsewhere... or people start starving.