The flux capacitor is powered by Mr. Fusion. You can see it on the back of the DeLorean in the following photograph. We can assume Doc put some appropriate items into it (such as food and other materials like at the end of Back to the Future or beginning of Back to the Future Part II) and they then just need to get the DeLorean up to 88MPH which is done using a locomotive.
Doc: "You mean we're out of gas?"
Marty: "Yeah, no big deal, we got Mr. Fusion, right?"
Doc: "Mr. Fusion powers the time circuits and the flux capacitor. But the internal combustion engine runs on ordinary
gasoline; it always has. There's not going to be a gas station around
here until some time in the next century. Without gasoline, we can't
get the DeLorean up to 88 miles per hour."