Who should review
The person responsible for the HR in the business should lead the review, as they’re the person who will know the most when it comes to the ‘what’ needs reviewing.
The people who use the system the most, your employees, should provide input too, as they are - obviously - best-placed to comment on employee experience.
The person in charge of the money should get involved because if you do need to switch to a new provider and there are budget constraints, they might be more perceptive to your business case if they’ve already understood the reasons you need to switch.