The 8920A manual mentions an IEEE-488 interface available as a factory option, but I can't find any details anywhere.
It'd completely change the project, but it looks like you could read the digital data from the DMM using the J601 connector, along with range info from J501. J601 output is BCD, with strobe lines. I don't see any connections that would allow control of the DMM. I'd guess the IEEE-488 interface is talk only?
And as others have mentioned, galvanic isolation is a good idea. The option for a frequency counter incorporated a pair of transformers for isolation.
(I don't have an 8920A, so don't have any reason to complete this project. However, I've done similar interfaces in the past for other manufacturer's equipment...)