“Original” vital records, such as marriage certificates, are owned by and kept at state or county records departments. For a few bucks, you can purchase certified copies, which act as originals. Certified copies generally have a raised county seal and wet signature, so they are not equivalent to photocopies.
When we got married, we ordered like 10 certified copies just to have on hand for immigration and other purposes.
It can take several weeks to receive them, depending on where you live. If you can’t get them before your departure, maybe have your husband send them to you via post.
My guess is that for Sweden, a photocopy would be adequate, but I’d have some certified copies as well, just in case.