Suggested edits are disabled on per-site metas, which means that only users with full edit privileges (>2k rep on graduated sites or >1000 rep on public betas) can edit posts on a site meta. This has been declared status-bydesign by Jeff Atwood a while ago, with the justification that another queue moderators and high-rep users would have to keep watch on would be a bad idea.
But as already mentioned in comments there, the ideal solution would be to just add the suggested edits on the meta to the queue on the main site. That way there would be no additional queue, and the meta sites would be editable by anyone like all the main sites.
The whole SE network trains its users to expect being able to edit any post, disabling that functionality silently on the meta sites breaks the user expectations and only leads to confusion. It also drastically reduces the number of users able to edit on site metas for many sites.
I don't know if there are any technical aspects that make this solution difficult to implement, but from an external perspective just adding the suggested edits to the main queue seems like a good solution to this problem.