Yes they do:

EF has a list of people who want Avios and other reward availability and checks to see if there is availability, by asking for it.

They are managing a list, and I'm sure optimising queries to check availability for everyone on the list as efficiently as they can.

BA (or any other airline) could do this too.

If I was doing it I would take the distinct set of dates and routes people want to travel on, query the availability on all, and then tell each individual the answer that I found. I (and colleagues in my company) definitely have the skills to do this, given a suitable availability information source to query.

(Whether they should optimise it in this way is a different question, but they could do it).