If they don't fly, they're not aphids. That would be my first guess as the nasty little suckers adore sunflowers. Can you post a close up?

They could be leaf footed plant bugs, lace bugs or others. Grab a magnifying glass and get a closer look. Insects go through several life stages and sometimes th young ones look very different from the adults.

Do a google search or follow the link below. Click on the 'true bug' section. Details and control methods are included.

Here is a link that might be useful: Pests of sunflowers

I have brown aphids on my Mexican Sunflowers (Tithonia rotundifolia). I had limited success spraying them with a castile soap and water mix (2 tsp. soap to 16 oz of purified water).
The aphids were on my Maximilian sunflowers, but the ants that tend them must have moved them to the Tithonia. Hopefully the beneficial insects will ramp up soon. I have seen a few assassin bugs, praying mantis, and loads of lacewings.
The aphids don't seem to be slowing the sunflowers down much so I'm done spraying, let nature take its course.

Sunflowers grow faster than even heavy infestation of insects can eat. Planting sunflowers near other crops may lower infestation levels of other crops.

Here's a picture of them that I took today. They are weird, they just hang out, I don't ever see them move!

Shape looks aphid like, you could just spray them off with a stream of water with a hose if you don't like the sight of them.

I agree aphids. Wash off with stream of water (best). Wash off with insect soap in mornings (if water doesn't remove most of them). Control ants with tanglefoot which protect and move them around and protect them from predators. This assumes the sunflowers have not already bloomed. If they have completed bloom then just remove the stalks and discard so ants don't rescue.

Here is a link that might be useful: soap

Looks like aphids.
They have a flying phase to lay eggs on new plants, then the new nodes don't fly.
I just noticed them on my sunflowers,too. I'm not going to spray because it's near my artichoke and the predator bugs that show up will keep aphids off the artichoke, which is also an aphid magnet.
Nasturtiums and radishes will do this too, in the cooler seasons. And cowpeas, in the heat of summer.

I just moved the sunflowers, as they were in a pot. Little bugs are still on them and appears that they're eating the little guys. Oh well, as long as they didn't get to my veggies I'm fine. :)

The aphids are gone, replaced by another pest the Western Leaf-footed bug.

This could be the culprit for some of the tiny critters on your veggies or flowers. Plant bugs go through several life stages and the young ones (nymphs) look very different from the adult.

Follow the link below for a close up and more info.

Here is a link that might be useful: plant bug nymphs and adults