The Southwest Chief runs between Chicago and Los Angeles, through the fabulous American West. Crossing Mississippi, through 8 states, canyons, ranches, deserts and mountains provide mesmerizing scenery that are not visible from interstate highways.
Few facts before jumping to the questions.

Previously named Super Chief

, the train/route was inaugurated in 1936.
Santa Fe(flagship train) merged with Super Chief(all-coacht train), being known as El Capitan, source Wiki.
The Southwest Chief route is 2,265 miles that cross through eight states in about 51 hours.
In certain places, the train reaches nearly 100 miles per hour and at other times, for example through the canyons, about 20 miles per hour.

Does Amtrak Southwest Chief Have WiFi?

However, to get to the subject, unfortunately there is officially no WiFi on the Chief.
Can I get connected to the internet while traveling in sleeper cars on Southwest Chief?
Some cars might have WiFi routers on them, but I wouldn’t count on it.
What are my options to connect to the internet during my trip?
Your best bet would be to hotspot your phone. You can learn how to create a mobile hotspot in the video below.

Another good tip would be to check your phone carrier coverage map.

Depending on your phone carriers, here is some great info on that:
If your provider is T-mobile, check your coverage map here.  For Verizon here, and for ATT here.