Every Appraisal for a rating is done by a team. The minimum number of people is 4 and that can include the Lead Appraiser. Every person on the team must meet certain individual pre-requisites and contribute to certain team-wide qualifications. It is best if the team's constituents include people from your company as well as outsiders.
At the appraisal, if you don't have (and can't create) qualified people in your company to be on the team, then you will need to bring in outside team members. (Most Lead Appraisers keep these in their back pockets -- kinda.) Outside team members are essentially consultants and charge as such. You're doing well if you can get outside team members for $1000/day. This would be very high-value. And, if you're only charged for a day where 1 day = the date on the calendar, and not 1 day = 8 hours, you're doing VERY well.
If your organization needs to get up to speed on CMMI, you'll probably do one of two things: (1) Look to hire an employee with the expected expertise, or (2) Look to hire a consultant with the expertise.
Which you choose to do depends on your organization's needs. The pros and cons of either approach are a basic matter of business and strategy. Either way, there's a cost. As for consultants, they're a lot like Lead Appraisers. And yes, many Lead Appraisers are also consultants. So, what and how they charge is largely up to them.