I know that Eyefi cards can use Wi-Fi networks to transfers photos. I also know that there's some sort of smartphone/tablet app that you can download to do various things with your Eyefi card and photos, but in the case that a Wi-Fi connection is not available, can the Eyefi card somehow link to a smartphone/tablet and use that smartphone/tablet's 3G/4G network to transfer the photos to a predesignated computer? Thank you.
You could turn your device into a Wifi hotspot and then EyeFi can connect to the internet via your phone (or tablet).
I don't have any experience with Eye Fi specifically, but presumably its programmed to scan for a list of known wifi hotspots. I would imagine you could program it to find your phone consistently and then once your device is enabled as a hotspot and as long as your phone has internet access to the target machine (ie. that target machine needs to be exposed to the www as a server of some sort) you'd be off to the races.
Of course enabling your device as a wifi hotspot will eat up data usage from your cell plan, so make sure you've covered that angle. If your plan doesn't already include data it can be a pretty sizable hit to the wallet.
A quick Google search turned up these pages:
From the specs I could not see any kind of 3G/4G compatibility, therefore I don't think EyeFi can be used with these kinds of networks. Should they decide to offer such a product, it would have to include a nanoSIM, a wi-fi transmitter and the memory itself, which is quite an effort in that tiny space. Anyway, no network in the whole world would support unlimited full-speed download/upload of huge amount of data (provided you shoot in jpeg maximum quality a photo is more or less 10 Mb), so I guess nowadays it would be useless to include a 3G/4G module in EyeFi cards...
EDIT: Sorry I misunderstood your question. If you can manage to make EyeFi selectively connect to a Wi-Fi network (I am sure that it doesn't only connect to open Wi-Fi, but you can set which one to connect to), you can of course use your tablet or mobile as a tethering hotspot. Basically the EyeFi would connect to your phone/tablet with Wi-Fi as if the last one it was a router. 3G/4G connection would then be used to link your EyeFi to the internet