The repair was pretty simple, just involves pulling out the oven, unscrewing some things, and swapping out the igniter.
First take the bottom liner out of the oven to expose the igniter and unscrew the 2 screws holding the igniter in. You will likely need some WD-40 or similar to get the screws loose.
Once that's done, pull out the oven and remove the screws from the back panel and take the back panel off. With the back panel off, you should be able to see the heat resistant cabling for the igniter coming out of a small box-like cover near the bottom of the back of the oven. There's 1 screw holding the "box" on. Remove the "box" cover and disconnect the connector to the igniter. Thread in the new igniter wire and plug it into the connector.
Screw everything back together, push the oven back in, and you're done!