Usually when people ask me about this, my recommendation is 'Don't.'
However, with the pandemic and a sudden surge of people sewing stuff, my local usually well stocked fabric store is pretty much bare. I couldn't shop local if I wanted to. So here's some tips on shopping online:
First, the more info a shop provides about the fabric the better. Ideally, you want:
- Picture of the fabric with a ruler for scale
- The fabric content/type
- The fabric weight
Here's a screencapture from Fabric.com. I'm not getting paid by them or anything, this is just a good example of a fabric listing that has everything I look for:
Now obviously you want to see the fabric before you buy it. I don't need to explain that. Some websites put a coin in for scale but that isn't really ideal. I'll hold up a ruler and enlarge the fabric on my screen until I get a good match to get an idea of whether the print will be in a good scale for dolls.
The second part is the fabric type. Most websites will have this info also. "Quilting Cotton" is my go-to fabric for most patterns because it's the most common/easily found fabric. Therefore, whenever possible, I make sure a pattern will work well with that weight.
BUT NOT ALL QUILTING COTTON IS EQUAL.
If you're sewing for a bigger doll, like a 1/4 or 1/3 scale, this next part is not as important, but if you go for 1/6 or smaller, it becomes super important, and I'll show you why:
These are the stats from two different fabrics in my last fabric order. Both are quilting cotton, both are in the 'lightweight' category, but Fabric A is about 20% lighter than Fabric B. Heavier weight means thicker fibers which means a stiffer fabric. I can use Fabric B for Barbie Dolls, but it only works well with fitted clothes, like a pencil skirt. Fabric A though, can be used in gathered clothes like a cupcake lolita skirt AS WELL AS fitted clothes.
Moving up to 1/4 scale, Fabric B could be used in a gathered skirt. It will still look stiffer than Fabric A in the same style, but it will work and not look wrong.
Get up to 1/3 scale, if I make the same gathered skirt out of both fabrics, Fabric A might need an underskirt to look full, but Fabric B won't.
If you can't find exact weight, look up the brand and fabric type and try and find it on a website that sells the same brand/type. Usually all the quilting cotton from Company A will be the same weight, the variation comes more between companies.
Outside of quilting cotton, there are a few materials that are consistently good for doll sewing. 
Both are cotton so share similar qualities with other cotton fabrics (non-stretch, will take a crease, doesn't fray easily) that makes it good to use for dolls, but if you look at the weight you can see they are even lighter than quilting cotton.
If you're looking to get a selection of solid fabrics to keep in your doll sewing stash, 'voile' is good. It's not expensive, and performs well in both gathered and fitted styles at all sizes. It comes in prints and solids and varies from sheer to opaque, and some of the higher end stuff has a nice sheen to it as well.
Gauze is a good fabric for clothes with softer looks. Quilting cotton and voile are tightly woven which makes them smooth and crisp. For styles that look lived in (ex boho/hippy type styles with flowing blouses and skirts) gauze will do well. Gauze also performs like a silky when cut on the bias (diagonal) as the loose weave will stretch over contours for a fitted cut without needing to sew darts.
For knits, go by the same rules.
Knits are for your soft, stretchy clothes: t-shirt, tunics, turtlenecks, leggings. Stick to things in the lightweight or very lightweight categories, and if the weight is given, lighter is generally better--unless you're going for a miniature sweater look!
Avoid silkies if at all possible.
Silkies are the hardest fabrics to pick out online because weight isn't necessarily related to stiffness, and it really helps to be able to see the cut edge to get an idea of how fray-prone the fabric is. For example, I just tried to find some light taffeta, but out of a full shelf of maybe 20 bolts of fabric that could all be described as 'light taffeta' only about 4 bolts were soft enough for doll clothes and low-fray enough for doll clothes.
Fortunately most doll clothes patterns that call for a silky can also use voile. Nice voile can even have a bit of a shine, so especially in the 1/6 and smaller sizes, voile will give you at least as good of a result and be much easier to work with.