TL;DR: Reactor doesn’t require Internet access to work, except for geofence conditions and some notification actions. Caveat: long-term loss of Internet breaks may break Vera’s clock and cause all automations (Reactor and otherwise) to malfunction. Read detail for more info and mitigation strategy.
Logic and automations are all standalone within Reactor except for those things which naturally require Vera’s cloud or other outside cloud service. This includes things like geofencing and notifications and should be no surprise.
There is a catch on the Vera side with wide-ranging effect, including into Reactor: Vera hardware doesn’t have a real time clock. That means it needs to synchronize time to external time servers, usually in the cloud. If your Vera is up and running and your Internet access drops out, that’s usually not a problem, for a while. After a little while (an hour or so), Vera may reboot, and that’s when things really come a gutser. If Vera reboots when there is no Internet access, it comes up with a default time and date (that is very wrong) and runs with it. This will make any absolute-time-based logic malfunction: if you have something that happens at 2pm every day, it may happen at what is really 1:30am because of the broken clock. If a task runs on Sunday, it may run on what is really Tuesday. When Internet access is restored, the clock will sync again — time will appear to leap forward, and any delay tasks will immediately fire because the delay period has gone through a time warp to expiration.
Reactor attempts to detect a bad clock and has facilities you can include in logic to warn you when Vera’s clock is invalid. There is also a setting you can set to just have Reactor not run at all if the clock is bad. The Reactor documentation describes all that. In any case, this poor clock handling affects all subsystems within the Vera, not just Reactor (if you have time-based native scenes, they also won’t work correctly, for example). At least with Reactor you can detect it and choose to do other things, or nothing at all.
If you routinely lose Internet access for long periods of time, I recommend that you change the clock source on your Vera. You can, for example, very cheaply add a hardware RTC to most cheap micros (RPi, ESP8266, etc., and some models already have it on board) and make it your local network’s time server — configure everything in the network to sync to it. Your mini-clock will sync to the cloud and everything else will sync to it; because the mini-server has a hardware clock, if the Internet drops out and it can’t reach cloud servers, the hardware clock continues with the last-known-good time until it can sync later (and it’s usually then only off a second or two), even if power fails (hardware RTCs have their own batteries, typically).