Using a third party is not illegal in itself. But you need an agreement with the third party as to how they will store/process any user data they collect.

This is fairly fundamental under GDPR. It's the 'data controller'/'data processor' split.

I suspect (but IANAL of course) that most CDNs would fail here, because the blanket agreements they offer are basically worthless.

But it's easy to imagine a CDN that has a different business model (charges a tiny amount pr. resource stored, for example), and is completely fine under the GDPR.