Not easily. There shouldn’t be anything stopping you from using Cloudflare DNS for your client’s domain. All DNS you’re currently using should migrate over to Cloudflare DNS so you can continue to use email, plus Cloudflare services for the website.
To avoid using Cloudflare DNS, you’d have to connect with someone with a higher level plan that can accept CNAMEs from external domains or a partial DNS setup. This is not a common setup unless your host is already a Cloudflare partner.
Thank you for this very helpful response. I’m really concerned about MX records being moved over properly. Is there an import process to move all hostrecord/other settings over or would I need to do this manually afterwards?
Cloudflare will attempt to import records for you, although you really need to verify the details manually. You can do all this before making any nameserver changes.