How to get into a phone that has Find my iPhone activated but no sim card? Or contact the owner of this lost phone?

I found this phone on the sidewalk today and I have been trying to find out who it belongs to. I thought that if I went to get the sim card, I could contact their provider (like ATT, Versizon, etc) and ask them to return the phone. But there isn't even a sim card on the phone. It also has this Find My iPhone thing, so it feels like there's nothing I can do. Any suggestions on how I could get it back to the owner?

Maybe taking it to Apple would help but I don't really know.

The obvious ways you return valuable items that aren't phones still work. You give it to the police for safekeeping (which also keeps scammers from convincing you it's theirs when it isn't) and you advertise locally (Craigslist, physical bulletin boards) saying you found a phone and giving contact information for yourself. You can have people describe the phone and its case, and refer anybody whose description matches to the cops.

Once "Lost Mode" is engaged (via "Find my iPhone"), unless the owner entered a message or any contact information at the time they engaged it… There's really not much you can do to return it to the proper owners.

The best course of action is to give it to the police & let them handle it. With the phone in "Lost Mode", nobody EXCEPT the rightful owner of the phone can disengage it & anybody who attempts to do a factory reset will run into the activation lock (which is part of "Lost Mode") that requires the original owner of the phone (which the phone will only provide minimal information that would be enough for the owner to ID the proper account) to login the phone before it'll proceed through the initial set-up. This practically turns the phone in question into the proverbial "iBrick" (rendering it useless to anybody else).

Assuming the rightful owner lives in your area (or at least has an idea where they may have lost their phone), they'll likely check with the police on such matters or may have submitted a report. Just don't try to sell it or keep it for yourself, as that will just lead to more headaches for you.