I've worked on feeding operations for both the Red Cross and Salvation Army. Keep in mind that things differ from state to state, chapter to chapter and even disaster to disaster, but here's how things have worked here.

Red Cross doesn't really have strong mobile cooking resources. They do have ERVs, their ambulance-sized mobile feeding units, but those don't have any cooking capability on-board. Each chapter maintains one for local use and to deploy to other areas. Locally we did mobile feeding maybe 1 or 2 times a month at larger fire scenes, mostly for fire-rescue and law enforcement, but some times for those affected by the disaster. We either purchase food from suppliers (local restaurants) or get food from a church or a charity like Metropolitan Ministries. Despite working multiple declared disasters, I've actually never seen any of the Red Cross mobile kitchen units, but I believe they have them.

The Salvation Army does much more mobile cooking. Each of their mobile feeding units is propane-equipped and can prepare hot food. (Like the Red Cross, I believe each local Salvation Army "core" maintains a mobile feeding unit). They also have mobile kitchens that are very impressive. I worked a search and rescue staging area the first 36 hours after one of the 2004 hurricanes and the Salvation Army brought in their kitchen (18 wheeler trailer) and essentially turned the keys over to the SAR task force. We ate well that night. The kitchen is pretty complete and includes a walk-in fridge and freezer and space for palettes of supplies.

I think if you really want to make an impact, I'd start by talking to your local Red Cross and Salvation Army chapters. Both really are great organizations with a lot of good people, despite the occasional bad apples and mis-steps they've made. One of the two is probably doing food service (canteens) for firefighters at large fire scenes, which is the most regular need.

If it isn't them, there's probably a local firefighter union that either has or wants to start some sort of mobile feeding operation for first responders. They can be a little closed off to a non-firefighter but they might welcome the help.

Most religious denominations have some sort of feeding capability but to be honest, it's usually pretty slow in arriving and doesn't have anywhere near the impact of Salvation Army or Red Cross.

One of the almost desperate needs most of those organizations have is for self-starter kinds of people that can talk to companies and get them to agree to make donations, especially long before any disaster strikes. With the Red Cross we constantly had to find new suppliers for all sorts of things (hotels, restaurants, clothing, wholesale food, etc) and get purchase/donation agreements in place. I'm still amazed at some of the wheeling and dealing I saw volunteers pull off. I had a volunteer that I could call at 10pm "I need dinner and snacks for 75 first responders" and within an hour we'd be somewhere loading in a solid meal (not just Big Macs), all for free.