If you equip the Master Sword, use the camera rune, and switch it to selfie mode, you can turn the camera just right to capture the sword. The image probably won't be of a very good quality since your body will be in the way, but it will count for completion. It is also possible to purchase the image for 100 rupees, as well as any other image in the game, by going
to the Hateno Tech Lab and talking to Symin.
The images purchased this way are usually a lot better than any I've taken.
Specific images can't be purchased. You pick a category, and it gives you a random image out of that category. You might need to purchase most of the ones in the weapon category before it gets down to the Master Sword. It is impossible to get duplicates, so you'll never waste money brute-forcing it this way. You can also reduce the amount you have to pay (and increase the chance of getting the Master Sword image sooner) by filling out most or all of the Weapon category before attempting the purchase.