Whenever you see a sauce separate, it's because you have an Emulsion , which is two or more immiscible liquids.
In cooking, these liquids are typically water and fat.
To stabilize an emulsion, you use an emulsifier .
The most common food emulsifier is lecithin , and the most common natural source of lecithin is egg yolk.
If you don't want the taste of egg or your food is not going to be cooked (i.e. a vinaigrette), then it you can actually go out and buy pure lecithin (soy lecithin is common to find).
As the wikipedia entry mentions, there are other natural emulsifiers such as honey and mustard, and often when you see recipes calling for mustard when it seems to be a strange ingredient to add (such as cheese sauce), the reason is to help stabilize the emulsion.
Additionally, the most common emulsifier used in packaged or processed foods is sodium stearoyl lactylate .
It sounds scarier than it is; you can buy it in the store just like lecithin.
cooking.stackexchange.com/users/41/Aaronut
Xanthan gum will also work very well for holding this kind of sauce together.
Somewhere around 0.2%-0.5% by weight should be right.
Shear it in with a blender for a good long time.
I'm a little concerned about the food safety of your sauce though.
Are you certain it will remain good for weeks as you are planning?
cooking.stackexchange.com/users/1393/Michael Natkin
We have found that blending the sauce for an extended period of time keeps it from separating...
I mean days of blending...
Just leave it in the blender and when time allows give it a good blending and then let it sit...
You will find after about 3 days of periodic blending the sauce will no longer separate.
cooking.stackexchange.com/users/39999/Hot Sauce
You'll probably need some kind of emulsifier.
For a long-term sauce like hot sauce, you're probably going to want an industrial strength one, and I don't really have many suggestions there...
I never make sauces where I can't just use mustard or egg yolks.
Or you can just try sticking the whole thing in the blender for a while; that'll do it with salad dressing (for example), and it might work for your sauce as well.
cooking.stackexchange.com/users/218/Satanicpuppy
i was thinking cornstarch would work in cream sauces for reheating in the microwave
cooking.stackexchange.com/users/5868/tony
Last fall I tried my hand at making a homemade hot sauce. Essentially: roast peppers, blend with vinegar, garlic, salt and other spices, put in jar for three weeks, strain and you have hot sauce.   The sauce was great, but after about a week of sitting in a jar in my fridge it separated and became rather unsightly. Of course, after shaking it up it returned to normal. Is there a way to keep a sauce like this from separating?  Thanks!
Keeping Hot Sauce From Separating
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cooking.stackexchange.com/users/296/Vecta
The sauce separates because you have an emulsion of two immiscible liquids. You can use an emulsifier like xanthan gum, or blend the sauce for an extended period of time. An emulsion needs an emulsifier to blend it. You can also try a long period of blending.
You have created an emulsion. Try using an emulsifier. Xanthan gum is worth a try. Blend it periodically for a long period of time.