I just built my first bike up, it's a total frankenbike and it's a lot of fun, but I'd like, just for fun, to see how light I can make it as I do mostly use it for practical commuting. The frame is a vintage carbon MTB frame so I'm already pretty ahead, but I had to use a steel fork for the time being. I need it to accept a 26" wheel, with a threadless straight steerer, 1-1/8" with horizontal pull rim brakes if possible. I don't necessarily need to go full carbon, especially since I don't seem to find any forks that are both carbon and rim brake compatible, but just something that's fair
If you want suspension, you will not find a light 26" fork with posts for brakes new. Best you will do new is low or maybe mid range. You only realistic option is a used fork (or donor bike). Good news is you can often pick up something like SID for next to nothing, although it will need servicing and may be too worn out to be worth fixing.
However, I presume you are after a rigid fork. The crown to axle becomes the main measurement for a frame designed for suspension. Fortunately, this means you will likely find a 650B to 700C fork with close to the correct measurements. Unfortunately, this means that the brake posts will be in the wrong place. You may be able to find brakes that are adjustable enough to make this work, running the 26", or you could run mullet (larger front that rear wheel).
Consider going to disc front brake. The weight penalty is minimal, and it opens a much bigger range of fork options, and no need to run mullet if you go to a 650B or 700C fork.
The specific combination of low-weight, 26", and rim brake actually has a market: trials bikes. You'll get something overkill sturdy and plenty of tyre clearance, which strictly speaking makes little sense for "practical commuting", but it'll certainly work. Tapered steerers are the norm now but straight models still available too.
As usual, if you want carbon, you'll have to dish out a lot of money – e.g. a Hashtagg Carbon costs £330. Aluminium is much more affordable, e.g. Jitsie Varial 400 for £170. That's not cheap for a rigid fork, but it'll last forever and is still very light.
Especially if you intend to ride at least some off-road, I definitely wouldn't insist on 26" and rim brakes. Discs are better for almost all use cases, and I'm sure you'll find something used with Shimano hydraulics for a good price/value. A 27.5" front will most likely work just fine on that bike as long as it doesn't have too long travel. Light 27.5" air suspension forks are available for cross-country, but they're expensive and require a lot of maintenance. If you want suspension on a commuter bike then a cheap coil fork makes more sense, but these are much heavier.
Perhaps you'll also find a 26" short-travel fork in the dirt-jump category.