Yes, but it depends on the card generation and your setup. Older Eye‑Fi cards can be used on an existing Wi‑Fi network and upload to a server you run yourself. One reported approach is running a Python-based Eye‑Fi server on a Synology NAS, which lets the card transfer images directly to the NAS over your network.

A key limitation is that newer Eye‑Fi Mobi cards were noted as working only in their own ad hoc/direct-connect style workflow, so they are not suitable if you specifically want the card to join your existing network and copy to a share.

So if you want fully automated camera-to-NAS transfer without tethering, the viable path mentioned is:

1. use an older Eye‑Fi card, and
2. run compatible Eye‑Fi server software on the Synology NAS.

If your card only supports its own direct wireless connection, it won’t meet your requirement for transferring through your existing Wi‑Fi to a network share.