Looks to me it's this bulb 5 watt lw scientific bulb . Also found a this one on Ali express it's looks the same but I'm not sure aliexpress link

As it is a Fluorescent bulb you probably can't just retrofit leds to the contact as there maybe is a starter for the bulb. Probably best way is to use a multimeter and take some measurements around the lighting circuit(please be careful as this needs to be done when it's on). You then probably need a some transformer to get lower voltage and DC.

Does the bulb or fitting carry any markings to identify what it is?

The base looks like a Type S, as far as replacing it with an LED in a DIY fashion the leads are plain old 110v AC. You don't want to put much more than 5w of LED in there, so you can probably get away with a basic capacitive dropper circuit that matches the LED you choose. Someone mentioned a starter, that is in the bulb base, not the socket on the microscope on these.

There are LEDs designed to run off fluorescent ballasts but usually for more standard sizes (like T5). You could search around, no clue what size bulb that is, maybe something exists if you can id the bulb type

Otherwise you could pull the ballast and replace it with a led ballast that runs off 110/220 (whatever your locale uses). These are cheap and easy to wire. Then whatever bulb fits in the space. Just search bulbs before you get a ballast to make sure you get a ballast that outputs the correct voltage

Or just replace the bulb, easiest fix