I’m taking a gander at Wisconsin election law — as is my wont — and it looks like Ryan could appear on state ballots as both a House candidate and vice-presidential candidate:
8.03 Multiple nominations. (1) The name of any candidate who is nominated to the same office by more than one party or pri- mary or nominated for more than one partisan or state nonpartisan office shall appear under the party first nominating him or her or under the office to which he or she was first nominated. If the double nomination is simultaneous, the candidate who is nomi- nated, before the deadline for filing nomination papers shall file a written statement with the same person with whom he or she files nomination papers stating the person’s party or office preference. If the candidate fails to select the party or office, the filing officer shall place the candidate’s name on the ballot under either party or office, but may not permit it to appear more than once. If a can- didate is nominated at a primary election for partisan office or nonpartisan state office on a ballot where his or her name appears or by nomination papers filed by the candidate, and is also nomi- nated by write−in votes at the primary election to another office, or to the same office as the candidate of a different party, the candi- date does not have a choice, but shall be placed on the ballot for the election under the office and party for which the candidate’s name appeared on the primary ballot or for which the candidate had filed nomination papers.
(2) Subsection (1) shall not apply when a candidate for the office of president or vice president of the United States is nomi- nated for another elective office during the same election. If the candidate is elected president or vice president of the United States such election shall void the candidate’s election to any other office. A special election shall be held to fill any office vacated under this subsection.
Crowd, tell me if I’m reading that wrong.