I've enlarged the picture but I can't make out anything which looks like chickweed. If you can post some close ups maybe we can id the plants for you. It makes a difference to how you would deal with them Also I can't see much grass - does it go dormant where you are? All the time the grass is so thin weeds will take advantage of the bare round.

If plants are unwanted -- i.e. they're weeds -- the least that MUST be done is to remove the plants before they set seed.

See this info for chickweed: chickweed

Vinegar is a poor method to get rid of dandelions.
Vinegar only burns off top growth whereas the fleshy rot remains and grows again.

See this: dandelion info