Yes, it can do this with the DAQ6510 and a multiplexer card like 7701.
The instrument supports scans of channels where a measure function is assigned to each channel in the scan.
Scan supports the use of the Diode Test function where we force the selected amount of current and measure V using 10VDC range.
This use of Scan approach typically gives best throughput and easy to setup.
An alternate approach is to command a channel or channels to close. And then separately send commands to the internal DMM to config and perform measurements. Take note that closure of backplane relays to connect to the DMM is required as well as MUX channel closure.
In the case of model 7701 and Diode Test function, close "channel" relay 35 to connect card channels to internal DMM.
The first approach using scan takes care of backplane relays for you according to the assigned measure function.
See Multiple-Channel operation in the Reference Manual for DAQ6510.