I have a budget, made in excel, I have a 365 business and a team site used as document library.
Can I somehow let other users log in with "normal" emails and use these files, or are they prohibited from taking part?
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You can share sites externally with people not affiliated with your tenant/domain in Office 365. It has to be allowed by your global administrators and your site collection has to be enabled for it.
You simply share the site, library, or file with an external email address. That user will get a notification that it has been shared with them. Depending on the sharing configuration of the tenant, they will either have anonymous access to the shared item or be required to log in with an existing O365 account or Microsoft account.