I searched for "macintosh vga adapter" and almost all were exactly what you need. They look something like this:
15 pins on both sides, one is the 3-row, high-density 15-pin like normal VGA, the other side is the looser 15-pin configuration like is on the back of your LC.
With an old Mac like the LC, your problem is more complicated than finding a hardware interface adapter. The original LC and LC II had a maximum VRAM of 512 KB, which was adequate to support the old Apple 12" or 13" RGB displays, but not the multi-sync displays that followed. The LC III had a little more VRAM, maxing out at 768 KB, but still has the same limitation as its predecessors. A modern 19" LCD display typically has a widescreen aspect ratio (16:9) and an optimal resolution of 1440x900 @ 60 Hz. The LCs don't support this resolution, because it didn't exist back in 1990. Even in newer (old) Macs with PCI slots, there are early graphics cards that can't support widescreen displays, because they weren't programmed for it. If you want to use the LC, you should look for an older Apple RGB display, but because they had the Sony Trinitron CRT, they're very heavy and not feasible to buy from an eBay seller, when you factor in the cost of shipping. If there are any thrift stores or computer recyclers in your area, you might want to check them out.