Does BusyContacts support file attachments?

iCloud limits the total size of a contact card to 256k, and that includes the profile photo. So, in most cases, you will not be able to add an attachment to a contact unless it is very small.

Google accounts are also known to not support attachments.

As a work around, you can drag an attachment onto a contact card and that will create a link to the attachment.

Yet another option would be to instead upload these documents to Google Drive / Dropbox / One Drive / Any-3rd-party-cloud-storage and then paste a generated / shared link to this document in the contact's notes field.