What is the best way to keep weeds from overtaking the vegetable garden.

I think hay is a good choice. That's what i do with my grass clippings. It also retains moisture for your veggies.

If it's a vacant part of the plot, then weighing down black plastic or old carpet with bricks will surfice; keeps out the light and moisture. Having said, you can't beat regular work with a hoe especially when the sun shines.

Assuming that the veg. garden will be planted I wouldn't use a mulch at all. Firstly, clean the plot as well as possible before any sowing or planting. Grow the crops in straight lines with a decent distance between them then run the hoe down the rows regularly whilst the weeds are still small seedings.

If you exclude all the light under your plastic the weeds will die eventually as light is essential for their growth. If you use hay to mulch the weeds will just grow through it unless its extremely thick. It sounds as though you planted before getting the perennial weeds out. This needs to be done before you plant anything. Sorry there are no satisfactory short cuts.

I like a good couple of inches of hay between the rows - retains moisture, erosion control, deters weeds and you don't have to step in the mud on rainy days.