Who should train business stakeholders
very short answer: Someone who has the knowledge
and a hint upfront: It is not only that they should be trained, they must be motivated to do the "new" part of their work and as always with "new" work, nobody wants to earn new workload.
Now the longer and explanatory part:
I have no idea which framework you start to introduce and why you need change the existing workflow. I assume you want to introduce something more agile and leave the pure waterfall way, that said only to understand my thoughts.
From my experience it is important that:
every one in the organization has to understand that all disciplines have to understand a new framework and must accept that they are part of it.
that especially in agile frameworks the customer has its role in that framework which means attendance in framework events in a much higher frequency as in waterfall models.
any try to introduce a framework only in parts ( of the framework itself ) or only in parts of all effected disciplines must fail
This opens the point, that not only the "who" trains someone is important, but more that co-work is absolutely required to make the new framework a success story. That means, that all involved persons have not only to understand what the new tasks for them are but more important, how all the work of all disciplines is interacting. For agile frameworks it has one important part: It is not longer a chain workflow as putting something from the top into the queue and waiting on a product at the end of the cycle, it is now an interactive workflow, where on every team and role interface a feedback loop is introduced.
It is wise to give the stakeholders also the perspective that the total amount of work will not increase, because a lot of (old) work will be reduced. This increase the first acceptance rate :-)
Back to the "who" should do it:
My experience is, that external persons can be react more flexible on questions and have a higher acceptance rate by introducing something "new". Stakeholders are typically feeling "higher" in hierarchy, it is maybe not a good idea that "lower" disciplines try to train them. That is not a question of qualification but a question of acceptance! And it did not help, that upper management tell someone, that "you have to" do something. This may increase the "wall against the new" instead.
My hint: Take some external guide which helps you to setup the new framework over all disciplines even if it requires a bit of money. Let these guys help you all by taking part of all your workframe events, especially by starting new projects or during general introduction step.
In later phases you already have a good chance to improve your workflow by taking all the events set up for this topic. All modern frameworks have the "inspect & adapt" sessions, which can increase interdisciplinary understanding and lets your daily work move forward more softly. This will especially help people which are not have really "hands on the product" but keep customer relations which is also very important.
Some remark on the comment:
It sounds bad to keep old behavior on the track of moving to agile. "Waterfall deliverables" is a perfect reason to fail I belief. Why you need feedback to get opportunities? Who is deciding which workflow you practice? If the organization decides to move to agile, it is not about waiting on getting opportunities. If you see that your current business did not drive well because waterfall did what it always did, you have already good arguments to move forward. And if you decide to do it: Do it! But don't keep staying on the half of the way. That will waste all your resources and all the motivations. There is no "half of agile". You can't earn the fruits if you don't do a full step. Hanging in the mid is collecting the bad of both sides. Don't do it!