^ Yes absolutely. If the PSU stays on then you can insert multimeter probes into the back of connectors, but most probes may need a needle attachment to get down into the plug or to pierce insulation if access from the back into the plug isn't reasonable. At that point, you would replicate the usage scenario where the system was turning off and take the readings... not necessarily trying for the highest load the system can manage but rather the least load that is allowing the problem to be seen, especially for the following reason.
If the PSU is running well enough to stay turned on but failing (out of spec) under load, then the odds are you have bad capacitors and shouldn't really be subjecting the system to that, as it is the most dangerous time for the system if the PSU stays running but isn't working properly.
For this reason I don't recommend load testing if you feel the PSU performance is off, until after either:
A) Swapping in a different PSU if the one in question is under warranty still.
B) If warranty is expired, open it and check for vented capacitors. If caps are vented there's no point in doing the load tests, it needs replaced or repaired.
C) Spend the money (or DIY build...) on an independent load testing device. It need not be very fancy, just some connectors, switches, power resistors, heatsinks and fans would suffice. This is more in the realm of professional reviewers or electronics repair personnel, not computer shop level, modular component replacement type repair work.
As far as other load testing goes if the PSU won't stay turned on, you can consult the PSU spec sheet to see what the minimum load specification is. Often it may be around an amp or two on the 5V and 12V rails and for some fancier server PSU, also on the 3.3V rail. If you have a bunch of old HDDs lying around they can be used to cause that load.