You can search for award availability on ITA, but it's complicated. I found this article which details how to search using fare codes, but only for one airline is it written. I found this thread looking for an already put-together guide on the rest of the airlines, but theoretically, it's the same process,m just learning to use the airport codes, and fare codes.
Well, I can't post links yet apparently, but it's from thrifty traveler, and here's a bit from the article:
A Next-Level Tip to Find the Right Fares
How on earth can you find those L, U, T, X, and V fares you need to use these certificates? Are you supposed to just search route after route on Delta until you finally come up with a result?
There’s a tool that can easily help you zero in on flights that work: It’s called ITA Matrix. I’ll show you exactly how it can help in this situation.
Think of ITA Matrix like Google Flights on steroids … and then some. While it’s a fairly advanced platform, it has a ton of powerful features that you won’t find on Google Flights or other sites. Most importantly, it allows you to search for specific fare classes … including the ones you need in order to find flights to use your Delta companion certificate.
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This article is about finding an available companion ticket availability, but the concept is the same.
Moderator's edit: Here is a link to the article:
Struggling to Use Your Delta Companion Ticket? Read This Guide