Can bookkeepers make payments on clients behalf?

Well, yes you could. The question is whether they would allow you to, or you want to go to the nause of becoming an authorised signatory for each client. As a single company it is much easier to set up a pay run with one input and one authorised signatory (who uses a code to authorise the BACs batch). For you to set up one for each client would require access across multiple platforms.