Just contact your municipal clerk/office (township, village, city) and request information about the process and how to do it.
No excuse required to vote absentee in Wisconsin.
If you are worried about the mail, you can also drop your ballot off at the polls on election day IN-PERSON or hand deliver it to the municipal clerk at any time before election day. No one can do it for you. YOU MUST deliver it yourself.
You can also register at the My Vote link in the OP through this week, I believe. You may still need to bring proof of residency documents to the polls on election day.
Don't forget, you can register in person in WI on election day (or at your municipal clerk's office) up until the day before the election. Just bring your proof of 30 day-residency. If you recently moved, you are required to vote at the polls covering your previous residence. Proof of residency can include your vehicle registration documents - which most voters have if they drive to the polls. Don't let anyone turn you away because your driver's license has a different address.