I am wondering if I can replace this type of bulb in this type of light fixture with an LED equivalent, or if I need a new light fixture altogether.
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Looks to be a bog-stock halogen bulb, and yes, there are LED versions.
Measure the length before shopping. You'll want it in millimeters, generally.
Bulb shape is T3 or R7S as best I can figure. A good reference for those I have not found, but those seem to get product results when combined with LED Bulb
That is an R7S bulb. From wikipedia:
An R7S is a double ended, recessed single contact (RSC) linear halogen lamp measuring either 118 mm or 78 mm. Some less common lengths are 189 mm, 254 mm and 331 mm. These lamps have a T3 shape on an RSC/R7S base. These can also be known as J type and T type lamps.
You'll also see them called T3 or J-Type halogen bulbs - though I'm pretty sure those definitions are supposed to refer to the lamp holder, not the bulb. Searching for some combination of "R7S", "T3" and/or "J type" should find suitable replacements.
They come in different lengths - most commmonly 78 and 118mm, so be sure to find the right size. They also come in different brightnesses - some of those halogen bulbs are 300W or more, used where you want extremely bright lights. Keep that in mind when finding LED replacements - some will be a lot dimmer than what you removed.
There are plenty of suitable LED replacements. Stores have all kinds of styles and brightnesses of them.