Thankfully, this shouldn’t be complicated.
For job seekers, the only issue switching from one staffing supplier to another would be if you are currently under contract through the firm in question. Contractors typically can’t switch firms until the current contract ends. If you are not on a contracted assignment and are having issues with a staffing organization not providing you adequate service, you can easily find and switch to a recruitment company that better serves your needs.
For hiring managers, the only issue switching from one staffing supplier to another is if you want to keep your existing contract-based staff obtained through the firm in question. Due to legal agreements, that option isn’t normally doable. Therefore, you’d have to start using another staffing supplier for future hires and move on from the troublesome supplier as their current contractors finish their assignments. If the above situation doesn’t apply to you, then switching to another supplier is fairly straightforward.