I don't know about WebDAV but it does let you mount SMB shares.
If you need WebDAV I think Documents by Readdle will do it, and maybe even makes the share available to other apps though its File Provider. But I'm not sure what parts of Documents are subscription based now since I have a paid copy from before it went subscription.
It even has the option to keep a local copy of folders from network shares so you can use them offline, which is a pretty handy feature if you're working with large files or don't have a cellular iPad.
