So you seem to understand 肩身が狭い is an idiom. Apart from this, the remaining part of the sentence is a simple causative expression. に marks the causee, and は is a topic marker. Also note that ～思いをする is another set phrase.
We will make you have a feeling of "肩身が狭い" (and I feel sorry about it).
Although not explicitly stated, a sense of apology is usually implied in a sentence said by a noble person like this.
One definition of 肩身が狭い is "feeling ashamed", but it's broadly used to describe any situation that makes you feel as if you can't hold your head up high in society. In this case, the visitors (そなたら) might not have done anything shameful in particular, but the exclusive atmosphere of the village makes them feel somewhat constrained or unwelcome, and that's what 肩身が狭い is referring to.