I don't know of anything ubiquitous off-hand in the kitchen that is both high in potassium and ready to use on plants without composting or some other such. Kelp is supposed to have some. So, maybe if you have kelp in the kitchen, it could work. If you have any potassium bicarbonate (used for leavening) in your kitchen, that might work, but I don't know how much plants like it as a potassium source, and I don't know if it's organic approved.
If banana peels are high in potassium like Jacob Knight says, then if you're in a hurry and can't compost (and have indoor plants), I imagine you could incinerate some banana peels and put some of the ashes on your plants.
Probably the easiest way to add organic potassium to the soil is to use potassium sulfate (at least some potassium sulfate is organic approved, by the way, even though it's water soluble and a mineral chelate like a lot of stuff that isn't organic approved). The problem with potassium sulfate is that it's water soluble (which might not be much of a problem if you have more clay in your soil, or if your plants are in containers, but it could leach through regular land soil, I imagine, instead of adding potassium in your soil long-term). I used potassium sulfate a lot last year. It's effective (and works quite well with tomatoes). Look for the certified organic stuff if you want to be sure it's the organic kind. Purity levels seem to differ depending on where you buy it.
You can use wood ash for potassium, but it is also high in calcium and can raise your soil PH. You don't want to use too much, or it could be disastrous. Also, I've heard that wood tends to collect heavy metals from the soil. So, it's likely that wood ash could be high in them, although if you're only recirculating wood grown on your own land every time, it probably won't alter the levels much, unless the heavy metals come from way down low.
Greensand has a little bit of potassium in it. It is supposed to be a long-term solution, but not much of a solution, since it doesn't have much potassium. I'm sure you can safely use a lot more greensand than you can use wood ash, however. Greensand has other touted benefits (like improving soil structure and such).
Granite dust is a rockdust that is also supposed to add a small amount of potassium to the soil. I'm not sure how much or how effective it is.
Both greensand and granite dust will add certain other minerals to the soil, too.
You can get potassium from compost, but odds are, you'll have a lot of nitrogen in it, too. I understand nitrogen and potassium compete with each other and help to balance each other. So, compost will likely help a lot, but most of the benefit will likely not be a potassium benefit (even though there should be a potassium benefit). I imagine adding wood chips to your compost would decrease the nitrogen and increase the potassium, if you needed that. Wood chips are, of course, made from wood, which is what wood ash comes from (and hence must contain potassium). However, wood takes nitrogen to decompose. So, it should lessen the nitrogen in the compost.
Some potassium magnesium sulfate (Sul-Po-Mag) is organic approved and may also be a good source of potassium for you (but it also adds sulfur and magnesium). To be fair, potassium sulfate also adds sulfur. This is a quick-release form.
