When I had turkeys, they were pretty seasonal layers. Their laying season was definitely shorter than a chicken's. I would say April to November is fairly accurate. I'm sure with artificial light and a bit of heat, you could increase the laying time
Several people on the century of turkey talk thread( including me) have turkeys laying this time of year. They're said to only lay in spring, but some of my turkeys laid through summer and now some of my hybrid turkey hens are laying with no supplemental lighting!
I've got a few broad-breasted turkeys that I can't bring myself to kill yet. They were hatched in April of this year. They started laying a couple of months ago, and they are still laying strong, I'd say 2 eggs in 3 days each. I was surprised when they started laying since I'd read everywhere that turkeys are seasonal layers and won't start laying till spring. My BBB's both lay, and try to go broody, but they are fairly easily discouraged. It could be due to the fact that they are broad-breasted, I am not sure how consistently heritage turkeys would lay.
Another thing that surprised me about my turkeys is that they are laying now in the middle of winter without any supplemental light on fairly simple feed. I feed them a mix of cracked corn, oats, barley, black oil sunflower seeds, and some oyster shell. We still have enough grass, so their feed amount is pretty limited, enough to last them for maybe an hour in the morning, and they spend the rest of the day foraging on whatever winter grass they find. I give them some scratch in the afternoon, and that's it. When I tried to feed the same feed to my laying hens (with the addition of hard-boiled eggs and vitamins in water) they all quit laying. For a few weeks there I only had eggs from my turkeys until I understood my mistake and put my chickens back on layer feed, and now they all are back in business. Plus I use supplemental light for a couple of hours in the morning for my chickens. So, my chickens need a lot more encouraging to lay this time of the year, than my BBB's.