No wimpy and laying down isn't leggy. Leggy plants are tall and spindly, but upright and stretched up toward the light, because they aren't getting enough light. Laying down is probably "damped off." It's a fungal condition to which little seedlings are vulnerable, especially in conditions of high humidity and moisture and low air circulation.
I'm going to guess you started your seedlings in peat pots, maybe with a greenhouse dome? Get rid of all that stuff!
Type seed starting into the Search the Forum feature, and read some of the tons that's been written here about it, complete with pictures of people's seed starting operations.
No you don't need $70 grow lights, but yes lots of seedlings do better with light. Regular shop light fixtures (around $20) with regular fluorescent tubes (around $4 each) work just fine. And it's a one time expense. If you buy it once, you can use it every year.
But at this point, at least for broccoli and onions, you might want to give up on the seeds for this year and buy plants from the local nursery. The broccoli is cold weather plant. I start my broccoli seeds in late jan or early feb for planting out in March. Once the hot weather comes, they tend to fizzle. Broccoli started too late tends to go straight to flowers without setting heads, if it is already getting hot. The onions, if you want bulb onions (as opposed to green onions, chives), the seeds are best planted in the fall. they will over winter and then make bulbs the following year.