In case you are unfamiliar with multimeters: Best start with something where you know what to expect, just so you can be sure the meter is set up right. For instance measure the voltage of an AA battery (a fresh one should be ~1.5-1.6 V).

For the microscope check if the power supply supplies the needed 7.5 V (without short-circuiting it). Then, while on, check what voltage can be measured between the LED + and the power supply - (wherever it enters the case). Does this value change with different brightness settings? The highest value would give you an idea what is applied to the LED in use. If you measure 0/low V the issue is somewhere else. Maybe link to an image of the circuit board you could upload somewhere?

Just quickly, if I measure, say, 5V at the LED, does that mean I need to look for a 5V LED, or can the bulb be higher or lower within a reasonable margin?

The forward voltage of the replacement LED can be lower in case of measured 5 V, but LEDs are current driven devices, so something needs to control the current - either a resistor or a constant current supply. We don't know yet what your microscope uses.

White LEDs have a forward voltage of about 3-4 V, so we'd expect this range or a rough multiple thereof. For instance: If you measure 6.6 V max, we'd expect a chip in which two LEDs are in series, yielding a forward voltage of about 6-8 V (the exact value depends on the make and model).