Do vegetables ripen like fruit?
Vegetables don't ripen. Carrots, potatoes, leafy greens, etc...

They will get bigger of course and some veggies will get more bitter or woody with age. Greens will get bitter.

Generally speaking veggies are better young. Carrots are particularly sweet and tender when young. Potatoes are also perfectly fine at any size.

The possible exceptions would be the fruits that are labelled vegetables for cooking purposes like some squashes and tomatoes. These do need to be ripe and whether they can be ripened after picking will depend on the plant.