I think budgies tend to be winter/spring breeders but can breed other times of the year as well (some of the confusion may be due to some sources being based in the northern hemisphere).
But wild budgies don't really follow specific times of year - they mostly breed during the rainy season when there's more immature seeds, veggies, and other soft foods for babies.
Typically budgie show breeders will get their birds into breeding condition by providing much more soaked or sprouted seeds, fresh greens, veggies and eggfood, starting a few weeks to a month before breeding. It's good to provide smaller amounts of veggies and greens year-round, since they are healthy for budgies - but much more before breeding. They also need to be given plenty of calcium from a cuttlebone, crushed sterilized eggshells, or mineral block.
There is no season that is riskier for breeding. It's more about them being on a healthy diet before breeding and while feeding chicks. Also, overbreeding of females should be avoided for their health - but they are known to sometimes kick their babies out of the nest a little early so they can nest again. They need to be watched for this and if mom is picking on the babies at all, she might need to be removed from the cage (but kept nearby) and dad will usually finish caring for the babies.
Make sure you have good homes and time to care for the babies! Ideally, you will want to handle the babies while they are still in the nest so they grow up calm around people. Babies raised like this are much easier to tame, and easier to find good homes for.