Is a multimeter even the right solution here? Also would you be able to wire up a standard DMM and dynamically switch between current and voltage readings thus reducing the meters required to two. This probably depends upon your meter and test requirements.
A multichannel A to D card in a PC may be a smarter choice. It depends of course on how much of this testing you will be doing. If you get a card with 4 independent channels you can grab your data synchronously.
Scanner cards have their uses but can your test requirements handle the offset between the four readings? Only you can answer this question. With 4 multimeters you could trigger readings to happen at the same time.
By the way there is another option that is sorta half way between a bench DVM and an A/D card. That is to use something like Keysights USB connected DVM module. You should be able to put 4 or those in a small rack or use a scanner in the same rack. For example:
https://www.keysight.com/en/pdx-2905759-pn-U2741A/usb-modular-digital-multimeter-5-digit?cc=US&lc=eng. You can also look into VXI but that ups your cost a bit.
Another option is a Raspberry PI with some A/D boards. With the right libraries writing software here will be easier than trying to run a scanner or collection of DMM. We are talking low end for the most part but the selection of cards is large, see this list:
https://elinux.org/RPi_Expansion_Boards. Actually you don't even need to be focused on PI here as other SBC platforms have similar capabilities. The nice thing with this approach is that you can end up with a very nice test stand that easily sits on the network.
Finally don't dismiss using a Digilent Analog Discovery as a data collection device.
In the end there are so many options here that you can spend weeks trying to choose the best option. Further the best option might be a combo of techniques. Given that buying multimeters and scanners would be my last option