Yes, you should be able to accomplish this by federating your ASP.NET application with Office365 directly. As you probably already know, typically the way this works is you have Office365 synced to on prem AD, and you have a trust set up with the on-prem ADFS server. When you're signing in to your web application the user gets redirected to Office365 to type in their UPN (typically the email address). Office365 uses that to figure out which on-prem ADFS server to redirect you to.
If you're logging on from within the domain, you get authenticated right away via windows integrated auth. ADFS will redirect you back to Office365 to establish a session, and Office365 will log you in to the application itself. If you're outside of the domain, you'll need an ADFS external proxy set up. There, instead of windows integrated auth, this special ADFS proxy will prompt the user for corporate credentials, and then redirect back to Office365 in the same way as before.
Here's a nice whitepaper that explains all of this in more detail:
http://www.microsoft.com/download/en/details.aspx?id=28971