Nebraska is a “no-excuse” state, meaning any registered voter may request an early-voting (a.k.a. absentee, mail-in) ballot, and is not required to provide a reason.
This year due to COVID-19, the Nebraska Secretary of State mailed every registered voter, except those who had already requested one, an early ballot application form. If you requested a ballot before the mailing and still received an application, don't worry, it wasn't lost nor do you need to submit an application again. It is more likely, it didn't process before the mailing went out.
If you still need an application here are the forms:
Douglas (English / Spanish) and Lancaster counties use their own forms. Most of the rest of the counties you this generic form (English / Spanish).
Applications must be physically signed. You can take a picture or scan your request and email it to your county election official. You can also mail or, if you still live in the '80s, fax your application to your county election office.
Check the status of your early voting ballot on Votercheck.