The edit queue being full is only one of many reasons you may be blocked from suggesting edits. Note that in addition to the site-wide limit described here, there is a limit onof how many suggested edits an individual user can have pending at any given time (5 on most sites, 20 on beta sites).
Once you have a good case for increasing the queue size, start a discussion on your site's corresponding meta-site site explaining your case. Allow time for the community to discuss your arguments and determine if a change is actually appropriate for your site.
Once it looks like it has a positive consensus, escalate it to Stack Exchange staff by tagging it with status-review if you are a moderator, or flagging it for moderator attention requesting for the tag to be added if you're not. This will bringraise the post to the attention of Community Management, who will take a look as soon as time allows.