Letterpress prints on uncoated 100% cotton paper and even handmade paper. Traditional card stock is not recommended for letterpress printing as it is too thin.
The thicker the paper, the deeper the impression can go without bruising the back of the paper. Printing is always a delicate balance of getting a noticeable impression without destroying the back of the paper.
If you are able, opt for double thick paper to get the best result. A cost effective option is to print the main invitation on double thick paper and the rest of the pieces on single thick (or even digitally print).