Can you find Pokémon without moving?

It is sort of possible, but not very efficient

You can try placing the app in "Battery saver" mode. This causes your avatar to wander around a little bit due to GPS measurement inaccuracies, even if you don't move. The wandering can trigger very close Pokemon and even allow you to spin pokestops when you wander into their range.

The movement is fairly random, so wandering will not be as efficient as intentionally moving around based on the nearby pokemon tracker. It also counts for hatching eggs but is very slow with that too.

It'a possible if you use nothing but Incense but would be very costly, not at all worth it in any way.

I have the app running while I'm at home watching TV and not moving. In one day I've caught Nidoran, Spearow, Rattata, Weepinbell, Gastly, Pidgey, Staryu, Weedle, Psyduck, Exeggcute and Venonat by sitting still. So yes, you can catch them doing nothing. Note: for some (not all) of them, I fired off an Incense.

If you're going to do this, just pop the app open every five minutes or so -- minimizes battery drain. Whenever you get the white spin of death, just close and reopen the app to reset it. Pokemon will wander across your path occasionally, and it's easy to catch a few this way when you're stuck inside.

Of course, you should figure out how far your internet extends, so you can do a little circuit when you can to somewhat extend your catching area. The more area wifi you have access to, to better, but you should look into purchasing a pre-paid data plan so you can go out into the wild.

This is only anecdotal, but my avatar seems to wander much more when I have bluetooth turned on than when I turn it off. I keep location services on in highly accurate mode, to use a combination of wi-fi and GPS, so it may be coincidental.

There are a couple of apartment complexes behind my office, and several folks in the office have wi-fi on their phones, so wi-fi location finding may be causing the wander. But it does seem to happen more with Bluetooth on. She wanders around the entire office building, and back and forth between two of the apartment buildings behind me. I'm leveling up and hatching eggs much faster now, but I'm also out of pokeballs because there are zero nearby stops for me to load back up.

In short, yes, you can catch Pokemon without moving.

However, if you are in an area that does not frequently spawn Pokemon, you will have a hard time catching anything good or on a regular basis. If you want to catch a lot of things with minimal effort, drive to a park or mall and sit somewhere (bench, coffee shop, etc). You will see much better spawns than you would at your house (assuming you don't live in the middle of the downtown area of a major city).

That said, sometimes rare Pokemon randomly spawn in very odd places. I leave Pokemon Go running all day at work and I have caught a Snorlax and multiple Scyther without getting up, though that is over the course of 2+ months.

Really, it's all dependent on where you choose not to move from.

A side note: the Pokemon that you see as being "Nearby" (now "Sighted") are somewhere within 200 meters of you. There is currently no way to know how close or what direction the Pokemon are in, though. The mall is probably your best bet for lowest effort and greatest reward, since there will be numerous spawns all around you.