For any secure connection (in your case WebDAV over SSL/TLS) the certificate store needs to be initialized. There are certificates in your personal key store which have their private key on a smart card. That's the reason why the window pops up. You need the card reader attached with the appropriate smart card or you get rid of the certificates in the store.
I am not convinced by your explanation, as I regularly use smart cards (EID, Bank, ...), there are 3 active in the personal store (6 certificates : 1 authentication cert and 1 signature cert for each). When I do not put a card in the reader, I get the warning (smart card login) 6 times. when I insert one of the active cards into the reader, the warning appears only 4 times. Since I do not want to remove the certificates, in any case I would get at least 4 times that totally useless warning.
If the connection webdav needs the certificate stored on a smart card, the store needs to be initialized, okay but, are you sure that there is no possibility to check if the certificates found in the store are not necessary for the desired connection and so bypass the warning.
Some other programs that also use a WebDAV connection over SSL does not have that problem.