This answer is based on the assumption that Scrum is the right framework for you. Without knowing your organization, your culture, and the details of this project, I can't say if it is for sure or not. Scrum is designed to provide "a framework within which people can address complex adaptive problems, while productively and creatively delivering products of the highest possible value".
If you have chosen Scrum, the next step is to organize the team into a Scrum Team. The teams should generally be self-organizing, to the extent possible. There are only three roles in Scrum - Product Owner, Development Team, and Scrum Master.
The Product Owner needs to be a person who can make decisions about needs and impacts of the work being done. It's creating the requirements, then ordering them and working with the Development Team to understand them. This is a single person (although I also favor having an alternate who can step in if immediate decisions are needed). Essentially, this person gives a single voice to all of the stakeholders of the project and represents them on the Scrum Team.
If this is a project to deliver organizational processes, I'd argue that someone at the executive level is the right person for this job. They don't need a complete understanding of the details of every stakeholder. They do need the ability to be a single voice for the Development Team. They may need to meet with various other stakeholders and work to resolve conflicts in priorities between work. Ultimately, they are accountable for making the decision on what work will add the most value.
If you don't have someone who is responsible for organizational processes now, you may want to considering opening a position and promoting or hiring into it. There was an interesting post on LinkedIn recently about the "Czar of Bad Systems" that I think underscores the need for someone to be responsible for processes across the organization. If you have requirements for a process framework (think CMMI for software) or quality management system (think ISO 9000 or an industry-specific variant), this person should have expertise in that. Background in or a desire to learn process improvement methods in general (Lean, Six Sigma, TPM, TQM) would also be helpful.
The Development Team is a group of three to nine individuals who have all of the skills required to deliver the work. This group is self-organizing and there are no sub-teams. There is simply one team that is responsible for taking the required work and getting it to a done state. It's likely that different people will have different sets of knowledge or skills, but the team as a whole is accountable and responsible for delivering the work at the end of every Sprint.
In my experiences, the best people to understand and create processes are the people who do the day-to-day work. This is true in any industry, from engineering to manufacturing to office administration. There may be requirements levied upon the work that need to be considered, but if the people who do the work also own the processes, those processes will be more likely to be accepted and more likely to be of a higher quality.
I do think that it is a mistake to say that you "don't have any established business processes". You may not have documented processes. You may not have consistent processes. You may not have robust processes. But work is getting done. People are doing something.
I would recommend training the staff, at all levels - executive, management, and line staff - in Lean. Depending on your industry and environment, there may be a specific tailoring of Lean done. I think that Lean pairs very well with the Agile methods - there's plenty of overlap in the underlying principles.
I would think that training people internally to form your Development Team is the right option here. You don't need to start with an expensive formal training for the staff, either, if that isn't feasible in the time and budget. Train a few people really well and have them come back to help.
Finally, the Scrum Master helps to coach the team. This individual helps with the techniques used by the team to understand the value behind the work they are doing as well as find and remove impediments and waste in the process. They serve the Product Owner, the Development Team, and the organization.
Although some people advise against it, I find it helpful to have one person on the Development Team serve as the Scrum Master. Although it does reduce their capacity to work, someone on the team is, in my opinion, in the best person to work with the Product Owner to make sure the team has what they need and the team's limits are understood, to work with the organization to ensure that all stakeholders understand what the team is doing and how the team is doing it, and with the Development Team to identify and remove impediments.
There may be room for a temporary consultant to come in and work with the organization to train everyone on various Scrum roles and responsibilities for an extended period of time (6-12 months).
