Anyone know of any completely flavorless syrups?

A syrup is by definition a thick sweet liquid made using sugar. If you are looking to make a thick savory liquid, perhaps you want to look into thickening agents. Starches and plant-based gums are the most common thickening agents. Some examples include:

• Starches: arrowroot, cornstarch, potato starch, tapioca
• Plant-base gums: guar gum, xanthum gum, alginin

Each of these will have its own unique qualities so you might want to research further into which meets your need best.

The two thickener that will most likely result in a clear, "smooth" thickness that is slightly similar to a syrup would be likely gelatin or pectin. However both of these will be much thicker at cooler temperatures. So depending on your application you will need to alter the concentration. Meaning, if your savory liquid is suppose to be consumed at room temperature or cooler, make sure not to added too much of gelatin or pectin. Try to test the thickness of your savory solution by taking a small sample aside and chilling it to room temperature.

Achieving a syrup-like consistency necessitates something like a sugar. If you boil sugar and water together (simple syrup), you get that consistency. If you boiled it without the sugar, you'd end up with an empty pot after the water had boiled off, but would never see a change in the consistency beyond the water evaporating into steam.

It's unclear what you mean when you say you're trying to use this "syrup" to absorb other flavors. Certainly you can use a milder, "unflavored" syrup such as simple or agave in a savory preparation, as long as you tone down the inherent sweetness with fat, salt, or acid, but the use of a syrup is largely textural, and it doesn't seem to me that a "syrup" is really what you're looking for. A syrup is, in itself, a flavor component in addition to all its other qualities that I and others have mentioned.

I need this syrup to absorb other flavors as I'm trying to make something savory

Rather than a syrup, you seem to be looking for an extract or an infusion. Whether it's vanilla extract, chilli oil or lemon peel oil, they all absorb the flavour or aromas of the compounds without any sweetness.

The high temperatures used in baking can affect some of the flavour depending on what you use.

If the consistency of your final product needs approach a standard simple syrup as well you'd need to create a base liquid, flavour it with your extract and thicken it as Jay explains.

Perhaps chia seeds will work for you?

When you add water or other liquid, the seeds are transformed into a gelatinous mass. The gelatin will take on the taste of whatever you add to it.

A typical ratio is 9:1 (such as 3 cups of water to 1/3 cup chia seeds. To change the consistency, add more or less water (or whatever liquid you want to use).

Methyl cellulose (food grade wallpaper paste) is as flavourless and syrupy as you can get. It is used in many barcode products like salad dressing, and savoury sauces

$100 for a 5Kg bag, 20g makes a litre