In short: There is no easy solution, that is always on and will not need dedicated setup and extra steps. But I will elaborate common similar solutions. Some of them will not be available on you camera.

Cable Tethering

This is the most widespread use of quick photo transfer in the professional sector and is used by many studio photographers. You simply hook up your camera to a tethering capable software via usb and start shooting.

Some software will directly load the images into the editing process (Lightroom, Capture One) some might just dump the files into a folder (most of the camera's own software). You can then set-up any other software to upload, process or backup the data. But you will need a computer and a cable.

Tethering over WIFI

Some recent cameras included tethering over WIFI. So you either have the receiving end in the same WIFI network or use a ad-hoc network created by the camera to connect and transfer the data. This usually only works with the software provided by the camera manufacturer.

This is very limited by the bandwidth of the network and might be too slow if you want to transfer many raw images quickly, but it might be useful if you just want to transfer JPGs.

To my knowledge the canon 5d does not have that feature built in.

Tethering via dedicated solutions

There are solutions to provide tethering with better support. One of these is CamRanger. I have seen this used by a photographer to quickly transfer photos to an ipad, where they could be seen and processed further. This solution supports many canon cameras. However, it is rather expensive.

https://camranger.com/product/camranger-2/

WIFI SD Card

In the past there where WIFI SD cards available. I found one such under the name of eye-fi. I have never used one of these and cannot say if they are useful (or even still available).

https://www.youtube.com/watch?v=F-CyKSbeN-g

WIFI/ethernet FTP transfer

Some specialized sports cameras include an option to integrate the camera directly into a network via ethernet jack or WIFI and to connect to FTP servers to send the files over the net. This is usually only a feature in cameras that will be used in high volume photography or sports photography where speed is of the essence to quickly make photos available to image agencies. However this is probably the closest thing to what you had in mind.

Backup only

If the main concern is the backup part in the field, another option would be a a second set of cards and then quickly drop the current card in a device like a gnarbox which will automatically transfer the content to a hard drive while you use the second set of SD cards to continue shooting. Although this makes sure you have a local backup, it does not provide any of the automation that you described, you can access the files on the gnarbox from a phone. Note that the device is pricey.