Doorbell cameras that work without Wi-Fi

Some do. A doorbell camera that records to a microSD card or an indoor base station keeps recording without the internet, and a wired one still rings your chime. What goes is the alert on your phone. Which setup fits depends on which of three things you mean by "without Wi-Fi".

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An ethernet cable coiled at a wall plate beside a front door, ready to connect
Where a cable can be run, the disconnection problem ends for good instead of merely improving.

No internet at the property at all

Then you want a wired or wireless chime with no camera in it, or a camera that records to a card and is checked in person. Nothing will notify your phone, because notifying a phone needs a connection somewhere.

Internet exists but not at the door

This is the common case and it is not really this page. Extending the network is usually cheaper and better than buying a different class of product.

You want it off the internet on purpose

Local-only recording to a card or a hub, checked on a screen in the house. You give up remote alerts, and that trade is the whole decision.

You want it to work when the internet drops

Different again. Some cameras keep recording locally through an outage and sync afterward, so that is a feature to check on the exact model you are looking at.

Dual-lens wired doorbell outside a door with its chime plugged in indoors
Wired

eufy Video Doorbell S330, wired, dual camera

A wired dual-camera unit with a second lens aimed at the step. That is the fix for the parcel blind spot a single wide lens handles badly. Check the wiring before anything else. It wants 16 to 24 volts at 30VA. That is well above what an older bell transformer supplies, so a healthy 10VA unit still will not run it. A chime is included, and it does not work with a eufy HomeBase.

Battery doorbell outside a front door with its chime plugged in indoors
Battery

eufy Video Doorbell 2K, battery, with chime

Battery like the Ring, but it records to a 16GB card in the chime that comes with it. No monthly fee, which is the whole reason to look at it. The storage sitting indoors rather than in the doorbell is the part worth understanding: taking the doorbell does not take the footage. That included chime also sidesteps the wiring question that breaks most installs, and the doorbell will run wired if you would rather not recharge it.

Storage glyph for SanDisk 256GB High Endurance microSD card
Storage

SanDisk 256GB High Endurance microSD card

The card for a camera that records locally and is not a REOLINK. High-endurance cards are built for the duty a security camera puts on them, which is writing continuously and overwriting forever. That is a different job from holding photographs. Check your camera's maximum capacity before buying this size. A card larger than the model accepts is not a card that works slowly, it is one that does not mount at all.

Yes, within limits. Cameras that record to a microSD card or a base station inside the house keep recording with no internet connection, and a wired doorbell still rings its chime. Phone alerts, two-way talk and remote viewing need a network, so those stop.
Yes, and that is the easier case. Plenty of cameras save to a card or a local hub with no monthly fee and keep their app alerts, because they still use your home network.
No. A PoE doorbell takes power and data over one Ethernet cable, so it needs a cable run to the door but no Wi-Fi signal there.