Welcome to the forum.
Most basenjis go into heat only once a year (early fall to mid winter) and most have already had their litters and pups have homes. But it won't hurt to contact the breed clubs nearest to you and contact breeders to get on their waiting lists for next year. If you aren't wanting to show, there's always a possibility of puppies that don't work out for showing or other reasons, so you can let the breeders know if you're interested in being on a list for older pups.
I have seen a lot of web pages without updates in other breeds also. A few breeders told me that they even took down their internet pages because of harassment by animal rights people. Many have moved their activity to facebook, so you might look for them there. Also, local clubs often keep their lists more updated. And, breeders here also may know which breeders are nearer to you.
As for long distance, find a mentor in the breed that you trust to help guide you. If a perfect pup came up, your excellent dog history will help, but your chances are going to be higher if you've made personal contacts.
I came from Rottweilers, btw. Not knowing the breeders, but the welcoming and genuine support from the Basenji breeders was a wonderful change. You may have breeders who are iffy about placing with Dobermans, lol. Soon you'll be able to tell them it's your poor Dobermans that need protection. My first 2 used my male Rottie as their personal chew toy. We actually locked them inside so he could play in peace.