If you are spraying with a weedkiller bought from a GC, then you are less likely to kill them. It would be quicker and much cheaper to dig them out, also digging over the soil as you go. From now on, would be a great time to start on them as they are not in such an active mode of growth.

Definitely digging them out is the way to go if you want to grow veg there afterwards.

I find that the roots get shorter ever time you dig them out. You get out as much as possible of a massive "crown" of roots. The left-behind bits regrow, but have had to use up some of the root tissue left in the soil to make a new growing point and leaves, so are shorter and more possible to pull out. Just go over the ex-bramble patch every week or two for a few months, pulling out any attempts to grow again from the brambles. Mind you, you'll still have to keep an eye open for bramble seedings, but if you get them when an inch high, they're easy!

you could cut them right down if your not in a hurry and get that weed barrier membrane and cut out the light to the plant bye covering it completly if you dont have much elbow grease .you will be able to reuse the membrane afterwoulds know doubt and as moon grower so rightly said it allows the soil to do well while killing the weeds / plants you dont want .

Digging then out a bit at a time may take a while but it's still quicker than spraying. If you just get the main roots out you'll be able to start growing next year. Any further growth can either be pulled out or sprayed. Good for you for growing more veg.

Just a thought, If you really can't get the roots out, a membrain and raised beds on top of it will get you started and smother the brambles at the same time. Problem is you'll need soil for the raised beds.

and in time any burried will still come through!
so dig , dig and dig again. no real short cut sorry.

the membrane will stop it photosynthesizing seaburn