Cellphones aren't just about internet access - it can be realtime notifications, like a text message if a room suddenly becomes available at a shelter. Or if you're applying for a casual job, they might need your cellphone number so they can call you to come into work (eg if someone's off sick and they need someone else to come and take their place). Without a cellphone you could miss those kinds of opportunities.

And the best way to learn how to use technology is to play with devices in your own time & practice. I would think most jobs now assume smartphone familiarity & wouldn't provide training, could be an intimidating hurdle to someone who isn't familiar with them.