According to EyeFi's support article Add a Wireless Network, you can configure an EyeFi card to connect to an existing network, by inserting the card to a computer connected to the wi-fi network, and launching the Eye-Fi Center software on the card. This will allow you to tell the EyeFi card which network to connect to, and to provide it the network password.

NOTE that EyeFi was acquired by Ricoh in mid 2016, and posted notice that they were discontinuing several products, including the Eye-Fi Center software, because they were shutting down servers hosted by EyeFi that the configuration software connected to. If that' the case, it's possible that EyeFi cards may no longer work as noted above.

Toshiba FlashAir cards support connecting in Station Mode, allowing them to connect to existing wi-fi networks. To do this, you have to configure the card by inserting it into a computer, and edit the /SD_WLAN/CONFIG file on the card. This file contains configuration parameters, such the the FlashAir's network name that you can specify, the network to connect to (and credentials), etc.

According to EyeFi's Mobi Pro support article and discussion forums, you can configure a Mobi Pro to connect to existing networks (called Infrastructure Support), rather than your computer connecting to the Mobi's own network (Mobi Pro only; this is not possible with Mobi).

However, according to this discussion at the Mobi Pro support site, it appears the only way to connect to the Mobi Pro card that has joined a preexisting infrastructure network is to use the EyeFi Mobi Desktop software, which can only connect to a single Mobi Pro card at a time.

So with existing EyeFi Mobi Pro cards, you can connect to existing networks, but you will have to either manually get the photos from each card on the network using the Mobi Desktop software, or use the Mobi Cloud service (which is free for 1 year after purchase).

If you have or are using discontinued EyeFi X2 cards, according to EyeFi's support article Add a Wireless Network, you used to be able configure an EyeFi X2 card to connect to an existing network, by inserting the card to a computer connected to the wi-fi network, and launching the EyeFi Center software on the card.

After EyeFi was acquired by Ricoh in mid 2016, they posted notice that they were discontinuing several products, including the X2 cards and the associated EyeFi Center software, because they were shutting down servers hosted by EyeFi that the configuration software connected to. So even though you just want to get the photos off your own cards on your own network, the configuration software was disabled, effectively bricking any configuration of existing X2 cards.

Toshiba FlashAir cards support connecting in Station Mode, allowing them to connect to existing wi-fi networks. To do this, you have to configure the card by inserting it into a computer, and edit the /SD_WLAN/CONFIG file on the card. This file contains configuration parameters, such the the FlashAir's network name that you can specify, the network to connect to (and credentials), etc.