I am trying to figure out what type of connector this is
It is a 2×4 surface-mount female pin header with 1.5mm pitch.
and if I can buy matching parts to crimp them on my own wires to create an extension cable.
I couldn't find matching male pins anywhere. The only 1.5mm pitch connectors I found were single-row, or with specialized footprint.
This means you'd need to make your own.
Did you notice that the female header is only surface-mount to the board, the aluminium substrate board having a cutout for it?
This is what I'd do, because it would be reversible, but doable without any special tools or components: desolder that header from the board, soldering wires there instead, with a blob of high-temperature silicone (like gasket silicone) to keep the wires in place, well separated from the aluminium substrate board.
Gasket silicone you can get from any hardware or automotive store. Use RTV (room temperature vulcanizing) silicone with a platinum or tin catalyst; not an acidic one or with anti-mould reagents like in caulking silicones, since those may harm the board or components or the PVC insulation.
The other solution is to grab some 1.5mm pitch single-row connectors, for example JST ZH connectors (that you can get from e.g. TME), and modify them to work for you by making two 1×4 single-row connectors you can glue (plastic weld) together. Essentially, you'd need to file or sand one side thin enough so that when pushed together, the row spacing is just 1.5mm. Most of these connectors are PA66, i.e. nylon, with glass reinforcement; and for best results, you should use adhesives that are designed to bond Nylon 6,6 to itself. A combination of JB Weld PlasticWeld and CA glue is known to work, and hot glue actually bonds well with Nylon, but takes a bit of experimentation and practice to get right. Although, gluing them together isn't really necessary, especially if you use the 90-degree male connectors so the wires don't push against each other.